It depends.
I've got both.
I have had my bag get absolutely soaking wet, twice that I can remember. Both times it was a synthetic bag and while it was kind of a rough night afterwards each time, it was ok.
First time was in the mid eighties on a solo sheep hunt. A big wind came in while I was hunting, strong enough that it about knocked me down, and blew my dome tent away ! Found it in a creek 1/4 mile away, all busted up. Wrung the bag out and basically slept under the stars in it after that. Bag was warmish but soggy, hung it in the willows the next day and the wind dried it out surprisingly quickly.
Other time was just 3 years ago on a float trip. Apparently the zipper on my fancy new "dry" bag wasn't splash proof. Shocked at the end of the day to find that everything inside was wet. Same thing, wrung out the sleeping bag as best I could, made camp and went to bed.
Point being you are doing the right thing by thinking about it. Stuff happens.