The short answer is yes. Of course you can. I have allergies to cat dander, dust, and dust mites. But I'm not allergic to feathers. My allergist tested me to let me know what pets I could and couldn't have safely before I ever got my first bird. After looking at my results, he advised against owning a dog but said that I should do fine with birds.
Because of my dust allergies, I knew I'd want a relatively neat species that could live in easy-to-clean housing.
I decided on a Peachfronted Conure, which turned out to be the perfect bird for me. As a result, I've been a happy -- and sneeze-free -- bird owner for many years.
Sadly, I've heard from people who feel frustrated by what they say is overly conservative advice from their doctors. Some allergists will tell people not to own any pet -- even though their test results show that they're not actually allergic to feathers.