I had a very similar problem recently. Took me a while to chase it down. Turns out a cable with nothing connected had a VERY intermittent short and would quickly grab and release line 1. This was causing some grief since the customer would complain that callers were getting their calls answered and then hear static or get disconnected. They also noticed other lines would ring immediately when line 1 was idle Telco to site numerous times and found nothing wrong with which I tended to agree.
I had my new test set with voltage reading on line 1 monitoring and I noticed the display quickly cycle from -48V to -5V back to -48V. An indication that something quickly went off hook then back on hook. I saw this occur a couple times and only once saw the line indicator on the phone change to briefly on then off. Eureka! I knew something on the system was seizing the line now on to figuring out what.
I set up DSS keys for all extensions, had to wait a while for the trouble to manifest (it was very intermittent as explained). I then observed extension 20 would briefly light up then go off. Tested the cable and, sure enough, there was a high resistance short. Found that the cable was running through a floor trough and the edge of the trough cut up the cable jacket. Repaired the damage and trouble fixed.
All of the above is still valid but you may want to look into, and eliminate, this as cause.