For permission, could is more polite: Can I leave early? vs Could I leave early? Both are fine with colleagues. For possibility, can often points to something likely, could to something weaker.
It can get crowded on that train treats it as likely; could flags real uncertainty.
Same sentence, different message
Can I borrow your charger? You are asking a colleague directly, casual and fine between people who know each other. Could I borrow your charger? Same request, one step more polite, good with a manager or a client. Neither is wrong; could just adds formality.
The meeting can run long on Fridays. You are stating something you treat as likely, almost a regular fact. The meeting could run long on Fridays. You are flagging a weaker possibility, less certain. Choose could when you genuinely are not sure it will happen.
Both versions still work side by side. If Fridays truly do run long, can reads as honest familiarity; could reads as careful hedging, which can sound evasive if everyone knows the pattern.
Where the choice is made for you
When you ask for permission, both can and could work, and could is the more polite pick. For possibility, can suggests something likely, could something less certain.
When you ask, both can and could work, and could is the more polite pick. The exception comes when you answer. Permission is granted or refused with can, never could: Could I leave early? gets Yes, you can or No, you can't.
Yes, you could would sound odd, because could is for asking, not granting. If you are refusing rather than granting, can still does the job, but softening helps: I'm afraid you can't today, sorry sounds far kinder than a bare No, you can't. For the reply itself, the choice is made for you: permission is given and refused with can, however polite the question was.