It's all based on quality. If you can push the quality on NQ mats high enough you can +3 it. But you won't be able to push the quality on crab bow synth with NQ mats high enough for a +3. This is because of the rank of synth makes obtaining quality harder.
You need a minimum amount of quality to get to a tier of HQ, most people speculate that it's something like this:
+1 - 0+
+2 - 70+
+3 - 170+
But then I further speculate that each level will increase the chance of the HQ further, for example you've a higher chance of getting a +2 at 170 quality than at 70, and 370 will give you a higher chance of a +3 than 170. But nothing is guaranteed. The formula is almost certainly a sliding scale where increasing quality increases chance of +1/2/3s.
I find I HQ more from touch ups than the finished synth, which is why HQing anything that isn't a finished item is a pain in the ass. Touch ups still use your quality, I see it the same was as finishing the synth for the first time, just another roll on the HQ lottery.