It's the requirements that kind of bubble up from very far away in a large org that cause the most stress. Almost no one wants their reports or close colleagues to feel inundated and always scrambling but in a big org it's still really hard to mitigate when the originator of a request is a few teams away from the one that is made to feel like a bottleneck (actually there'll be a ton of people feeling like the bottleneck on any project which is by definition not true). Actually the originator might not even feel it's urgent it just feels a little more escalated all along the path.

If you firmly say "no" or "I can't" people will be a lot more accommodating than you think. Easier said than done in the actual situation but it seems almost always true.