Why is my cat still sniffling and can't breathe after vet visits?

Poor Spike! There are a few things that could be going on here. First, he may have a condition called chronic rhinitis - basically, this is a chronic upper respiratory infection that causing changes in the tissue and bone structure inside the nose, which makes symptoms very hard to get rid of even with proper antibiotic treatment. He could also have a nasal foreign body of some kind (like a piece of grass), a fungal infection like cryptococcus which wouldn't respond to antibiotics, or even a nasal tumor.

Since he's been on antibiotics twice now with no improvement, you have a couple of options. Ideally, it would be best for Spike to have some further diagnostic testing to try and figure out what's causing the chronic symptoms - this would likely involve culture and cytology of the nasal discharge, skull x-rays, and possibly rhinoscopy (looking inside the nose with a tiny camera) and biopsies of anything in the nasal cavity that looks suspicious. Referral to a specialist may be required for some of these tests.

Or, if you prefer not to pursue any additional testing right now, your vet should be able to talk to you about other treatment options that may help to relieve his symptoms. Many cats with this issue respond well to anti-inflammatory drugs, long courses of antibiotics (4-6 weeks), intranasal antibiotic/steroid drops, or even adjunct therapies like laser treatment or acupuncture. I would recommend asking your vet for his/her suggestions on the next step. If they don't have any other suggestions or don't want to talk with you about options, you can get a second opinion from another vet if needed.