I have found success in the 4 times I've fished during light rain/drizzle.
Small lake in NJ - warm and light rain on and off for the 8 hours I spent on the water. 23 largemouth. Several 2 pounders and a couple 3s. Two other boaters during this time. Also witnessed them catching over and over.
River in Upstate NY - 6 hours and on and off drizzle on a grey day. Several chain pickrel landed. One LMB.
Lake O trib - got drenched once on a hot day but no tstorms in the forecast. on and off drizzle after that. Landed 8 lb sheepshead and a small SMB. But that's it.
Small lake close to Georgian Bay - 2.5 hours and very light rain. Hooked 8 bass. Landed 6. Two of them were 2.5 lbs to 3.
These are the four times I recall fishing in rain. And didn't get skunked. And I've been skunked plenty of times. More often than not in mid-day sun. But if it's grey and a little bit of rain in mid-day, I like my chances.
So in my experience, yes I find those conditions favourable for fishing. But then again I'm typically fishing small waters for bass. And like Joel said, the barometer can tell you a lot as well. And wind also plays a part.
So many variables affect the appetite and behaviour of fish. We'll all spend the rest of our lives trying to unlock those secrets.