How many outlets can you have on one 15 amp circuit?
Yes.

Also, check your breaker. You can have as many outlets on a single circuit as you like*, but you have to be careful because you can only pull so much current from the circuit at a time. For example, don't expect to run an electric chainsaw and a vacuum at the same time off a single 15A circuit. If you have a 15A breaker attached which is likely the case, I would recommend not running any major appliances (Such as a Microwave or Window AC unit) off of it, since that will take up most of your juice, and one too many lamps in the other 17 outlets could trip your breaker. If you need to run a heavy appliance, just wire in another circuit, and don't drill a hole through the wires near the breaker box (Voice of experience).

*Some areas such as Belgium limit you to 8 outlets on a circuit, or you violate wiring code.