Of course future plans "break down". It depends on the field you're in, but in many cases it's not supposed to be a rigid plan that's considered "broken" if you don't adhere to it perfectly - other things will always come up. The idea is just to keep adjusting the plan and the priority list on the fly, to keep on top of everything new that comes up, and talk the priorities over with everyone involved. I know in my group everything usually takes much longer than originally planned (we're in science, it's sort of inevitable), so while my own tasks get delayed, it also turns out my coworkers don't need them done as soon as they thought they would, so everything works out.
I know what you mean about getting overwhelmed and procrastinating, though. When that happens, you really need to sit down with your priority list and work out a new plan so that you know what you should be doing next, rather than feeling like you should be doing everything. Again, talk to your supervisor and involved coworkers about the new schedule to make sure you have the priorities right, adjust them if you don't, or figure out a solution if it turns out there are too many things that need to get done too soon.
In the long term, my best solution to the overwhelmed procrastination problem is to go over and clean up my priorities list and schedule on a regular basis (about once a month for me, but that may be very different for other jobs). The important thing is to set up a schedule so that your priority cleanup happens before you get overwhelmed.
One more note - if you do your "priority cleanup" and talk to the people involved and it does turn out that they wanted more done than you can get done in that amount of time, and there's no easy re-prioritizing solution to the problem, try not to get defensive. Apologize, explain that you're working on your scheduling and on trying to figure out how many things to commit to, and ask for help solving the problem. You'll probably get it, if only because your coworkers want your scheduling to improve, and they probably all know how it feels to be in your position. If nothing else, next time someone comes to you for help, they'll have an easier time understanding when you tell them you can't do that because overcommitting just won't end well.
And last, do say no to people - it's an important skill, and even if they're not happy with the response in the moment, they'll appreciate the fact that when you do agree to what they ask, it actually gets done, rather than getting buried in a pile of other requests all of which you said no to. If you don't want to say no outright, tell them you'll think about it and check your current schedule, then get back to them with how long you think it will take and when you think it could get done given all the other things on your plate. In particular, if you're also working on a bunch of other tasks from the same person, be sure to tell them "I'm also working on X, and both X and the new request Y would take a week - which one of them would you rather get done first?" so that they're aware of (and part of) the prioritization process and they can't complain later that everything didn't get done at once.