In general it looks like getting into orbit was preferred whenever possible, but there are abort indications for a number of spacecraft systems problems.
From page 3-1 of the Apollo 11 Mission Rules document:
The launch will be aborted for the following reasons-- ...
B. CSM
Environmental
Loss of cabin and suit pressure
Loss of cabin pressure and suit circulation
Fire/smoke in CM
Loss of cabin pressure and O2 manifold leak
Electrical
Loss of 3 fuel cells and 1 battery
Uncontrollable shorted main bus
Loss of both AC buses during Mode I or Mode II
Propulsion
- Sustained leak or loss of He pressure (source or manifold) in both CM-RCS rings (Mode I only)
...
D. Team discretion will be used for---
Suit/cabin contamination
Medical problems
The Apollo 12 lightning strike incident came within a hair's breadth of hitting these abort conditions: according to Wikipedia, all three fuel cells went down, and one of the AC inverters (thus one AC bus) also was offline. Presumably, with the launcher still doing its job properly, they would have allowed at least a short time to troubleshoot before abort if both busses had gone down; it was about a full minute from the lightning strike until Pete Conrad threw the "SCE to AUX" switch that allowed the CSM to begin to recover.