In Season 1, Episode 3 ("The Fuzzy Boots Corollary"), it's revealed that Sheldon is allergic to cats, when Leonard Hofstadter (Johnny Galecki) brings one home after being rejected by Penny. Sheldon is opposed to the idea and fears the feline will affect his asthma.
However, in Season 4, Episode 3 ("The Zazzy Substitution"), Sheldon acquires a "clowder" of cats in an attempt to replace his ex-girlfriend Amy Farrah Fowler (Mayim Bialik). Leonard is stunned to see Sheldon acting so affectionately toward the cats, which he endearingly gives names, such as Dr. Robert Oppenheimer and Zazzles. He starts off with five cats, but by the time Leonard brings Sheldon's mother Mary (Laurie Metcalf) over to help, his new pets are scattered across his bedroom, including in the dresser drawers.
In this episode, Sheldon's Season 1 cat allergy is non-existent. This has led many fans to wonder if "The Big Bang Theory" creators forgot about this detail, or if this drastic shift was intentional. On Reddit, u/supbear said, "Allergies can come and go. I was horribly allergic to cats for about four years."
Given Sheldon's tendency to occasionally hide the truth, such as when he lied to Amy about being able drive so that he'd get chauffeured, it's possible Sheldon selfishly faked his allergy in Season 1 to prevent Leonard from getting a cat. It's also suspicious that Sheldon's go-to song to feel better when sick is "Soft Kitty," which is quite literally all about a "little ball of fur."