Yes, increased exposure can reduce the ellergic reaction that results in the itching and swelling. The very first time we are bitten by a mosquito, nothing happens. That triggers the immune system in a way that subsequent bytes cause a allergic response.
There are roughly two levels of "immunity" to mosquitos that in most people can be brought on by high exposure. The first level will itch for a some minutes, but then after about 20 minutes or so the itch is gone and so is the redness and swelling. In fact, it's hard to remember where exactly the moquito bite was. I know because my body now responds this way.
From what I've read, the second level works the same way but symptoms are gone in about 2 minutes instead of 20 minutes. I can't comment on this from personal experience.
Mosquitos used to bother me and cause itching, redness, and swelling of a small area around the bite for days, like they do for most people. About 25 years ago I was doing a 2 night bike trip with a friend from Wellfleet on Cape Cod back to home in north-central Massachusetts. It was summer and quite warm and we didn't have tents, just a sleeping bag each and a platic sheet in case it rained (fortunately it didn't). The first night the mosquitos were really annoying. I sortof tried to cover myself with the plastic, but that got really hot and sweaty, and I still had to breath of course. Eventually I fell asleep anyway.
The next morning I got up and all was fine, then I remembered the mosquitos the night before and noticed to my great surprise that I couldn't find a single mosquito bite anywhere on my face, which was exposed all night. Ever since then, I've had the 20 minute reaction instead of the usual several day reaction.
However, while this makes the mosquito bites themselves more tolerable, it does nothing against the pathogens the mosquitos might carry. Around here we have West Nile and Easter Equine Encephalitis. Those can be pretty serious, but Malaria, Yellow Fever, and Dengue Fever effect many more people around the world. So don't go around thinking it's OK to be bitten by mosquitos just because your body has a tolerable reaction to the mosquito bite itself.