You really need to check the fuel pressure. It is indeed possible for the pump to run but not put out sufficient pressure to run the engine. It could also run but not build enough pressure due to a bad FP regulator dumping the fuel right back into the tank, or even a really badly clogged fuel filter restricting the flow.

Checking pressure is pretty easy. The low-buck approach is to buy a cheapo 0-100 psi pressure gauge at the hardware store (the kind designed for water systems, with a 1/8" pipe fitting on it) which should set you back about 6-8 bucks or so. Then you locate the Schrader valve on the fuel rail, remove the valve core and clamp a short piece of fuel hose onto it with the pressure gauge clamped into the other end, turn on the key, listen for the fuel pump to build pressure and shut off (about 1-2 seconds) and check the gauge for pressure. You should have around 35 psi minimum in order for the engine to run, which should be around 40 or so with it actually running (if it did run...)