1996 Accent approx 65,000 miles Automatic trans

Alternator does not seem to be recharging the battery. Battery was replaced with a new battery. Car will run 1/1/2 -2 hours on short trips around town. Running time depends on lights being on. Car will not start or will stall out at that point. Tell tale dead battery clicking noise heard when you try to start. Recharge battery fully and car will run again until what I believe is the battery draining. Removed alternator and had bench tested, 3 tests in a row were run and each showed alternator as good. Test was done at a local NAPA store. My gut tells me its the alternator but test is making me think about other possible causes.

Appreciate your insight and assistance.