One of the most common reasons why a cat can't stop sneezing is an infection. Allergies or breathing in irritants are other reasons why a cat would sneeze. If a cat continues to sneeze, it's best to take the cat to a veterinarian to investigate the cause of the sneezing.
An upper respiratory infection, similar to a cold in humans, is the most likely reason why a cat would sneeze, especially if the animal came from a shelter. The feline herpes virus and feline calicivirus are viral infections that cats may contract from other cats. These conditions make cats more likely to develop upper respiratory infections that can cause sneezing.