As soon as someone talks about hash inserts being O(1) I assume that they are talking about the average case performance when your hash algorithm is working.

So do I (well, I try to avoid the term 'average' - in this case, I'd use 'expected'. 'amortized' is another term a layman may call 'average'). But I stop assuming that as soon as 'worst case' is mentioned. Or 'mathematical proof'.