Tested negative for chlamydia and gonnorrhea 2 months after condom broke is that a good time period to rule those out and not to worry?

You can be certain you are not infected at whatever site was tested (presumably a urine or urethral swab test), at the time you were tested. However, gonorrhea (and less often, chlamydia) can be cleared by the immune system in a few weeks. So in theory you could have been infected. Testing after a new exposure generally should be done within two weeks.

yes