It depends on the culture of the firm.  What you're describing sounds like a fairly large firm, with several levels of management - but also one in which higher levels of management seem to have a somewhat higher than typical level of interest in day-to-day staffing assignments.  In that situation I can see how you could ruffle the multiple levels of feathers by making this request.  

Having staff who "self-assign" can certainly lead to chaos, by side-stepping those who are tasked with staff management.  These people do a lot of work to properly staff all projects based on schedules, budgets, and all team members' strengths.  I do understand that you followed proper channels and didn't just attempt to move yourself to another project of your choosing, and that the project that you wanted to be moved off was on hold anyway - but even then firms tend to look at these things as precedent-setting (i.e. if they let you move upon request then others will want the same option.) It's often not possible to staff all projects based on employee preferences, and most employers expect employees to understand that. It may also depend heavily on how you framed your reasons.  For example, asking if it would be possible to get certain other types of experience that the original project isn't providing might be looked upon more favorably (especially while that project is on hold) than if your reasons are more self-serving or indicate that you might be a difficult or immature employee (such as saying you want to switch projects because you're bored, or because of problems with team members.)  

Does your firm have annual reviews?  If so, that's usually a better setting in which to discuss your long-range interests in project types and experience that you're hoping to get.  If there are no formal reviews then you might ask if you could schedule a meeting with your various levels of supervisors, to discuss these broad career goals and whether they would be amenable to addressing those.  But it's probably not a great idea to approach this issue by asking to be moved off particular in-progress projects.

How were you bitten?

5839 listed good points. The case with moving to other projects is not entirely up to you or the management always, but also to the PM/PA of that project team. Provided that there is capacity, they will first be asked if they want you in their team. Therefore, in my opinion, you could try to approach these people indirectly and first get them to know you and like you. Then you can strategically plan the transfer at the end of a project cycle (long term planning here) by voicing your request in a diplomatic way (diversifying your experience) during your annual review. 

I asked about the "bite" because I'm curious what the repercussions were.

It's interesting to read insights from those with lots of experience in large firms.  Coming from a small firm, on only a few occasions over the years did anyone officially ask for reassignment.  In those cases it was due to personality conflicts (normal people vs. narcissists), and where the rearrangement was attempted, it was always a pain in the neck for most involved.

What firm of scale wouldn't normally reassign staff once putting a project on hold at that phase? Clearly shenanigans.

Perhaps your request was initially rejected, because they thought you didn't have the necessary experience / background to work on that project?

Eventually, when you were moved to other projects were you able to pull your weight? I've seen this go wrong with people who are too ambitious and overestimate their ability.

Also, at what stage in your career are you and what sort of persecutions did you face? Please give us more details.

it's impossible to make any call based on what information the OP provided. my summary of the situation: a bunch of ordinary things happened, OP got anxious, and then... end of story.

there's nothing wrong with requesting to work on specific projects or even different teams. most companies will consider the request if it's presented reasonably. the answer will depend on work needs and how the skills and personality of the individual employee can fit into other projects. sometimes they can't.

in fact as you move up in your career you should be in occasional discussions with management on your work role and what projects you're looking to work on. this helps us assign roles - we usually like when someone can give clear guidance on what they want to contribute to because it let's managers know the individual will be more proactive in doing that work (unless they're just a needy and useless employee).

changing teams in a big company is normal and acceptable too. it doesn't need to indicate a personality issue; often it's a smart career move to develop relationships with a broader range of groups within the company and learn how different teams manage things slightly differently.

but all i get out of reading the OP story is he was very anxious about a totally normal request to discuss what he'll work on next, and he felt bad about how it went. without telling us why.

The specifics that the OP added about what happened help with the bigger picture:  he's a relative newbie, he was the only person left as the point person for a mostly stalled project, and he asked to be taken off of it because of how "problematic" he felt that project was.  It's not wrong or unheard of to request to work on certain projects or project types, or to have discussions about one's role in the firm.  But if you're the only person left on a difficult project, and you're a new guy, and things seem to be a little slow in this firm (based on this stalled project, and the other one he mentioned that doesn't seem to have fully ramped up yet), then by making this request there's a pretty good chance you're positioning yourself at the front of the line for when layoffs happen.  

Look at it from management's viewpoint:  there still needs to be someone to answer questions and go to meetings.  They picked you.  Because... presumably it's part of the job for which they hired you, and was a role for which they thought you were an appropriate choice.  You're disgruntled about it and ask to be moved to something better.  It's not going to do much for office morale or for your relationship with your new coworkers if they move the new guy to something more palatable and then move somebody with more tenure onto the sucky project, to continue what you bailed out of.  It's the kind of thing where sucking it up and sticking with the bad project all the way through might have given you some credit to spend on asking for a better next project - but begging off prematurely made you look easily replaceable and an obvious target.

Yeah it's hard to quit a project (without quitting the job) and have that come out looking ok.  Most people are inevitably going to get projects they wish they weren't on, from time to time.  Sometimes you can spin your discussion as suggesting that you might be very useful on a certain other project, because of your special skill in [thing that relates to that other project], or that you've always been extremely interested in [that other type of project] and in fact it's one of the reasons you picked this great firm...  

Once years ago I was stuck working on little university renovation projects for a repeat client, usually all by myself. I hated those projects and wanted to be moved off them. I got good results by scheduling a meeting with the firm principal to discuss goals, and saying that one of the main reasons I had picked that firm was its strong focus on flagship hospitality projects, and that I was hoping to get a chance to be on one of those project teams if that was feasible.  I never brought up anything about being dissatisfied with the projects I was currently on - I just used a little flattery, and language that put the decision in his hands and recognized that I understood it might not be possible instantaneously.  

There are obvious exceptions where you're going to need to have a talk about the project and its team specifically, for instance if the PM is harassing you or the client is using racial slurs or something egregious and personal like that, or you're being directed to draw things you know are dangerous and illegal. But barring that level of dysfunction, usually the best approach is bigger-picture, and focused on what you do want to do and how enthusiastic you are about it and how useful you will be when you get that chance - and not framed as a discussion of why you don't want to work on a troublesome project.

Never do that that, You'll be blacklisted as a shiet-disturber.

Better to leave for another office than request to be moved to a different project.

Not that the other office will be any more accommodating or better; same old shiet, everyone of them - you'll find that out sooner-or-later.

You can, however, ask to work on a project, but be careful in how you ask.

I agree with several of the points made above. You were probably marked for riddance even before the project went slow and that's eventually why you were the one left on it.

It's a shame there was not transparency from management, but usually it is possible to understand things by talking to other coworkers.

I am curious as to what your level was in the company, given your 10 years of experience?

I would not request being taken off a toxic project, but try to be put on the kind of projects that you do like instead. 

Don't force them to look for something else for you to do, but let them see how your drive and experience or enthusiasm would be of great benefit to that other project, etc blablabla. 

Let them come to the conclusion (with a little nudging) that moving you would be better for that other project rather than force them to move you away from the toxic one.