I don't know how you determined the battery is good; but it sure sounds to me like it isn't. The starter draws many times more current from the battery than any other electrical device on the car. For this reason; you cannot judge the battery's condition by how well it runs anything except the starter. But here's a test you can do: turn on the headlights, and then try the starter, while someone watches the headlight brightness. If the headlights go out or dim way down when the starter is activated; then there is not enough power in the battery. THE BATTERY CABLE CLAMPS MUST BE CLEAN AND TIGHTLY CONNECTED FOR THIS TEST TO BE VALID. MANY CABLE CLAMPS WILL LOOK GOOD TO THE EYE; BUT WILL HAVE ENOUGH INVISIBLE CORROSION ON THE INSIDE SURFACES OF THE CLAMPS OR ON THE OUTSIDE OF THE BATTERY POSTS TO PREVENT THE STARTER FROM RUNNING CONSISTENTLY. A TAPERED REAMER TYPE BATTERY CABLE SERVICE TOOL IS THE ONLY SURE WAY TO CLEAN CABLE CLAMPS AND BATTERY POSTS.
Now, if your battery is fairly new, it might be that the alternator has recently stopped charging the battery when the motor runs; which would then cause the battery to become discharged and weak. The battery will also run down from the current required to maintain the computer's memory, if the vehicle is stored for more than 1 or 2 months. If you think this might be the case, connect a battery charger to the battery and charge it overnight (assuming your charger supplies at least 6 amps). 1 or 2 amp trickle chargers cannot be used to recharge a car battery. Or, take the battery to a garage or a store that sells batteries, and have it charged and load tested. A voltmeter cannot be used to check the available energy in a battery; only a battery load tester can do this. The headlight/starter test is a crude but workable load test.
If the battery starts the motor after it is charged; drive the car to a garage or parts store and have the alternator output tested. If the alternator is good; the battery may have become discharged if a light or electrical accessory had been left on, if the brake light switch had stuck on; or if there is a short in the vehicle's electrical system.