Is It Possible to Be Allergic to Mosquito Bites?

No, it doesn't seem normal.

You might try applying citronella extract as a mosquito deterrent. You can buy the essential oil (not the candle) and mix it into your favorite organic lotion base...it works wonders...and smells great! Plus, no chemicals seep into your skin! (OFF! and other products are horrible for your body)

it may be possible that you are infact allergic to mosquito bites
just like some people are allergic to ant bites,bee and wasp stings

I was bitten by a mosquito yesterday on the eyebrow. When I woke up this morning my entire forhead was swollen. Also, my eye is slightly swollen. I did some research and I found this:

More severe reactions -- rather than the typical itchy red bump experienced by most people as a result of a mosquito bite -- occur less commonly. These may result in blistering rashes, bruises, or large areas of swelling at the bite sites. People who experience extremely large areas of swelling after a mosquito bite (such as swelling of most of an arm or leg, for example) have been dubbed as having "Skeeter Syndrome."

In rare situations, some people may experience anaphylaxis after being bitten by mosquitoes. Other people may have experienced whole body urticaria and angioedema (hives and swelling), or worsening of asthma symptoms after being bitten. Typically, these symptoms occur within minutes after a mosquito bite, compared to Skeeter Syndrome, which may take hours to days to occur.

Hope this helps you.