Some of the systems have a capacitor start/capacitor run setup that might include a relay to switch the relays. If you have a DMM (Digital Multi Meter) with a capacitor scale, it's easy enough to check the capacitors. Disconnect on of the posts completely, then short the cap with a screwdriver or jumper. Take a reading, short again, take another read with the meter leads reversed. Your readings should be within 10% of the nameplate on the cap. They're not expensive, try to keep the mfd rating of the replacement the same, voltage nameplate can be higher on the replacement, no problem. If there's a relay in the start gear, you can check resistance on the coil terminals (one must be disconnected) as a basic check. Some are inductance relays and are more difficult to DX, but a continuity/resistance test of the coil is basic and illuminating.
Don't overlook bad connections, as they can increase startup current. Check the terminal connections on the compressor. On a 3 ton, you're looking at current that is easily capable of burning a terminal, and a poor connection can add enough resistance to cause flakey startup issues.
2nd what Ski advised, check with a clamp ammeter to confirm breaker is up to the job, and also to see if the start current is over what the nameplate on the unit specifies.
