[alistairSH]

...ending a sprint with a couple of late-night panic sessions to get everything out of the door...

My first reaction is you are not sufficiently grooming your backlog. Stories are left too large, with too many unknowns. If you tend to have a small number of large tasks, this is likely the case.

That, or you just aren't very good at estimating. A large number of small tasks would indicate this problem.

In the second case, you need to circle back with the PO and discuss which items can push to the next sprint. And make a concerted effort to estimate better during grooming and planning sessions.