Can a bad timing chain cause my distributor to break?

Hello, It depends on what you mean by broken. If the distributor is constantly out of time, though you have timed it over and over, yes. If the distributor shaft is shearing, or the drive gears are becoming damaged, no. The timing chain will develop slack and rattle, and you really have no other choice but to replace it. With all that slack, it can jump time, or the chain will stay on the teeth but allow 1-2 degrees of timing change on the tight side. If your distributor is getting chewed up on the drive gears, time to replace the camshaft. If your distributor shaft is shearing, there is probably too much play on the shaft. Inspect and remediate as needed.