Technically if you're rebuilding with (all drives - 1), and problems arise on one of the remaining drives, even it's only a single bad sector, you're exceeding what RAID 5 is designed to handle. And in practice one bad sector often means, more than one bad sector.
I have had RAID controllers drop a drive because of a single bad sector. I can not imagine those will allow a rebuild to complete if a bad sector on one of the remaining drives surfaces.
I suppose it may depend on policy of the actual software/controller whether it continues or not. A stricter policy could mean the rebuild is aborted.
"During rebuilding the RAID driver reads every block on all the surviving drives. If it encounters any bit errors, the rebuilding operation is typically aborted. The RAID is basically in limbo. It may stay in degraded mode or it may go into total failure mode. If the RAID consists of a large number of drives with large capacity, the probability of rebuilding failure can be very high. For example if the probability for a 2TB hard drive to have at least one bit error is 1%, then the probability of having at least one error in 12x2TB hard drives is 11%." - source
"In the situation you describe (a faulty disk + some unreadable sectors on another disk) some enterprise RAID controllers will nuke the entire array on the grounds that its integrity is compromised and so the only safe action is to restore from backup.
Some other controllers (most notably from LSI) will instead puncture the array, marking some LBAs as unreadable but continuing with the rebuild. If the unreadable LBAs are on free space effectively no real data is lost, so this is the best scenario. If they affect already written data, some information (hopefully of little value) is inevitably lost." - source
For comparison, if we look at SSD drives, the more high end the SSD (enterprise grade) the less tolerant for any type of corruption the policy is at the firmware level. If it can not guarantee integrity of the data some (Intel) enterprise grade SSD's are programmed to 'brick' themselves. Philosophy is: either we deliver 100% intact data, if this can't be guaranteed then no data at all.
So, all I am trying to say is that there probably is no 'one size fits all' answer. It depends on the specific RAID controller (or software).
Bottom line: It depends.
Not all may be lost if the controller refuses a rebuild, if you clone/image the separate members, data recovery tools are often to virtually rebuild and array and allow you to recover the data.