Thanks for asking your most important question on Just Answer. How old is your cat? A URI in a cat is usually viral. Some veterinarians will use antibiotics to prevent a secondary bacterial infection. Since your cat got better on 7 days of Amoxicillin, it is likely a bacterial infection, but the antibiotics weren't given for a long enough course, or were not strong enough. For a URI in cats, I prefer to use Doxycycline since many of the URI's are actually similar to kennel cough in dogs (bordatella). I would talk to your vet about a longer course of antibiotics to clear the infection. Also, if persistent, a chest xray may be in order.
Please let me know if I can provide you with any further information. I hope your cat is feeling chipper soon.