Usually yes, but not at the same time: zinc can bind certain antibiotics (notably tetracyclines and fluoroquinolones) in your gut and reduce how much of the drug you absorb, so separate the doses by several hours and ask your pharmacist.
The concern is not a dangerous toxic reaction but an absorption interaction. According to the NIH Office of Dietary Supplements, quinolone (fluoroquinolone) antibiotics such as ciprofloxacin (Cipro) and tetracycline antibiotics can reduce the amount of both the zinc and the antibiotic that your body absorbs. Lowering the absorbed dose of the antibiotic can make it less effective, which risks treatment failure. This interaction is specific to certain antibiotic classes and is driven by timing, so it is largely avoided by spacing the two apart rather than taking them together.
Separate the doses: NIH/ODS advises taking the antibiotic at least 2 hours before, or 4 to 6 hours after, your zinc supplement; MedlinePlus's tetracycline labeling says to take zinc-containing products 2 hours before or after the dose. This applies to zinc-containing antacids and multivitamins too, not just standalone zinc. The same spacing logic applies to other multivalent minerals taken with these antibiotics (iron, calcium, magnesium), which MedlinePlus's tetracycline page also gives timing windows for. Not every antibiotic is affected, so don't assume — the safest move is to ask your pharmacist or prescriber whether your specific antibiotic interacts with zinc and how to time the doses. If you've been taking zinc at the same time as a tetracycline or fluoroquinolone and your infection isn't improving, contact your prescriber. Keep zinc within the recommended limits (the adult tolerable upper intake level is 40 mg/day) and tell your doctor and pharmacist about all supplements you take.
This is general reference, not medical advice, and not a guarantee of safety. Interactions depend on your doses, health conditions, and other medicines. Always confirm with your pharmacist or doctor before combining products, and follow the dosing on each label.