Yes this sounds just like the fuel pump relay problem that a number of us have. The relay can corrode inside and cause the exact symptoms you describe.
If you get stuck with it not starting, try opening the hood and tapping the black boxes found to the very very rear, underneath the base of the windshield (often behind largish easily removable black plastic covers).
If you're more DIY, pull the black one that's around the size of a cigarette packet. It has a load of male pins in the bottom (maybe 10-15). This is the blighter. Open it up (easy to do) and look at the connections to the large copper-coloured relays on it. They are often in bad condition. A simple resoldering will solve the problem.
Otherwise, pop to the dealer and get a new one, but if you don't check it as above you won't know its definitely the problem. Then you've got all sorts to start worrying about..