People have different situations in life and the range of options that you have as an employee can vary pretty dramatically. However, here's how I tend to cope with these situations (in order of preference):
1. If I don't get the time to do what I think is necessary, then I lobby to make sure that it's my decision about how much time it takes. You're not going to get handed this on a silver platter, but the main argument is that as a developer you are making a good salary. The person paying that salary should get the best value from that money. If you have to do your job in handcuffs, you're not going to be able to give the best value.
2. If I'm not in a political situation to do #1 (and you should probably try to make sure you are before you actually stick your neck out), then I will note the constraints I have to work under. Making a project succeed when everything goes exactly as you want it to is relatively easy (well, TBH, I think it's never actually easy, but you get my point). What makes a great developer is making the project succeed even when you don't have everything your own way. So find a way to succeed with the constraints that you have. If you fail, make sure to make it obvious which constraints are causing the most problems. Usually after you fail you can get some extra leeway on those issues. Work on the issues one at a time, most important first. Eventually you can usually promote yourself to option #1.
3. If I can't do #1 and there is some reason why #2 fails (and I can't fix it), then I will go elsewhere. No point in sticking around in a place where you can't succeed. Wish everybody great luck. Express remorse that you couldn't get the kind of success that everybody wanted. Thank people for the opportunities that you had.
4. If I can't go because I have no other job, or I need to stay because my visa says so, or my spouse will go crazy from stress if I switch jobs, or a million other possible reasons... I'll keep working and keep studying exactly what's causing the problems. It's easy to conclude, "Joe is breaking everything. I hate Joe" (I only picked the name Joe because to the best of my recollection, I've never worked with someone named Joe). But you're stuck and you might as well learn about the situation in more detail. Wait and watch. I hate waiting. Man, do I hate waiting. But eventually it will pay off. Opportunities are out there. Your job is to recognise them.
That's it. Don't lie. You'll only get found out and then make your job a hundred times harder. Don't get dragged into petty politics. The job is hard enough without drama. Play it straight and give everyone the best chance they have to understand what you are doing and cooperate with you. Don't undermine people. Just do your best. It's not always easy sailing, but learn how to navigate rough waters and you'll be fine.