I have owned a 91 Accord in the past and most of the accords from around that year that I have come across (in family, friends) have had this one same problem one time or the other. In my case sometimes the car started but sometimes the starter would just keep cranking the engine but there would be no ignition. My first thought was the Main relay as someone previously posted, but after replacing it I realized that didnt help in my case.

After researching further, it turned out that the distributer assembly (not just the cap but the whole thing including the one rod thing that goes into what it looks like the the head) the whole assembly had a manufacturing defect where the oil leaks into it, the solution was to replace the whole thing, I did it myself, it was very easy (IMO) and took care of the problem.

In my cousins case, the engine just suddenly stopped turning over. just cranked when attempting to start and would never have ignition. I suggested the same solution as my car, and bingo! that fixed it.

In one case however (a friend of mine), the local dealer just replaced the ignition coil (as the dealer claims) on his car and it took care of the problem.

Anyways, thats just my experience.