I know that there are a million and one things that could cause this, but it puts me in mind of a possible blocked fuel tank vent. As you drive, fuel leaves the tank but no air can enter, thus inducing an ever increasing vacuum, to the point at which the fuel pump can't overcome it. Leaving it for 15 mins is probably enough to allow this vacuum to bleed away.
Very easy diagnosis for this. Drive until you splutter to a stop, then simply remove your fuel filler cap - if there's a noticable "suck" when you remove the cap, then you've found the problem. I'm assuming that like most cars, the vent is actually in the cap.
Hope this helps, let us know how you get on.