They certainly can. The medical term is Conjunctivitis. If you suspect your cat has this, a vet visit is a good idea as he/she may need eye drops or ointment for it.
Ah right, I just read your other thread. If you're worried that your cat can get conjunctivitis from you, don't be. It's not zoonotic (a disease that can be passed from animals to humans.)
I thought Spot had an eye infection a few months ago, but it turned out to be an abscess located just above his eyelid. Any inflammation of the eye (or anywhere else) should be checked out.
George had pink eye last year - courtesy of the vet's office because no one in my family had it (no people or cats) so I'm sure he picked it up when he was in for his checkup. Anyway - it was easy to treat.
One of my gang has a blocked tear drainage duct and he tends to get conjunctivitis easily. He needs to learn to wash his paws before he sticks them in his eye. Oh, wait, he does.... Becky
Below is a link to help explain conjunctivitis in cats. One of my cats has had this recently. The ointment did clear it up and I am using the preservative-free drops in his eyes which helps a lot.