Here's a wiring diagram so you can check the connections:
http://www.telephonecollectors.info/...nnections/file
What is the manufacturer & model of VoIP ATA you're using? It could be that it's not putting out sufficient ring voltage. Other people have had that issue, and some ATAs have software settings for the ring voltage. Measure the ringing voltage on your line when it's called and let us know what it is.
Also, it could be a problem with the switch hook, as you surmised earlier. Carefully examine the operation of the mechanism visually to see if it's working reliably and without sticking. Also, use the wiring diagram to verify continuity, etc.
Maybe the ringer capacitor is the problem, but it's pretty rare.