Your battery is probably "sulfated". It can deliver a short, high current burst of energy, but it has no reserve capacity. Without a pretty sophisticated carbon pile load tester it's not possible to do an accurate test, but here's a field expedient work-around that I've used pretty successfully. Disable the ignition by pulling the coil wire out of the distributor cap and grounding it (or just don't turn the ignition switch on with a stomp start) and spin the starter for 15 seconds. Have a helper hold a set of voltmeter leads on the battery posts (not the terminals) during the test. At the end of 15 seconds with the starter still turning, the terminal voltage should be above 9.5 volts. If not, replace the battery.
There's a recovery method for a sulfated battery if you're willing to go to the effort, but it's a lengthy procedure with only about a 50% chance of success.
Jerry
