Most RIs in snakes are caused by bacteria that already live in the healthy snake, but do not cause a problem until the snake is under stress, such as too cold temperatures.
Most sniffly sneezy rats and mice are suffering from a myco flare-up. Myco is also something that is usually present in the healthy animal, but it will flare-up and show symptoms due to some sort of stress.
Whether or not myco is one of the organisms that can cause an RI in snakes, I am not sure, although I doubt it. However, even if it is, exposure from a sick rat is probably not likely to cause an issue for an otherwise healthy snake. And, if it did, exposure from a myco carrying rat that is showing no symptoms would also carry a risk of RI for the snake. If that were the case, we'd see a lot more RIs, because almost all of our feeders carry myco.
Bottom line: no, snakes don't catch RIs from sniffly sneezy rats.