Although many faults could cause the effects you describe, my best guess would be that the pressure switch is not operating for some reason.
When a combi boiler �fires up�, it switches on the fan to exhaust the burnt gases � to ensure the fan is operating, the boiler senses the pressure drop (causing the air to flow). If the pressure switch does not sense the required pressure differential, then the boiler will not supply gas to the main burner (and will shut down).
This could be due to a faulty fan, perished tubing to the pressure switch, air leaks within the boiler, a faulty control board, loose wiring, faulty pressure switch and more.
Best call in an engineer to fix it.