And the other two cup competitions? Anyway, I think you'll find the reason smaller clubs don't do better, is because most of them (including ourselves), seem to think the cups aren't important anymore. We went out of the league cup to a League One side for ****'s sake. And you just watch how many mid-table Prem teams go out to lower level opposition this season. It's a joke. The likes of us, Newcastle, West Ham, Swansea, Stoke, etc should be all about the cups. But we give up on them too easily, so no wonder the big clubs are sweeping them up every year now.
I'd say the same about the League Cup, as that's been dominated by the Big 4 recently plus Tottenham. Usually because they end up taking it seriously when it's the only trophy they have a chance of winning. I'd say on a good day you're not far off Tottenham's level (at least going on last season) so I'd say you have a much better chance than in the FA Cup, but again it's a case of it being possible, rather than realistic at the moment, especially as the semi-final is over two legs.
I honestly don't know enough about who's in the Europa League this season to make any informed judgements