I've heard people say that an engine will make more power after it is "broken in". Is this true? Is it making less than the rated power in it's first few hours?

This is generally a true statement. A brand new motor has small machining imperfections in the journals and cylinder walls. Once these have worked themselves out (this is the junk that you remove from your first oil change), the motor will spin more freely, suffer lower internal losses, and therefore produce slightly more power. There are several schools of thought on engine break-in. If you want a motor to produce the most power, but aren't concerned with it living more than a few races, go ahead and whale on it right out of the box. This will wear slightly more material off from the journals, resulting in a bit more bearing clearance, but less load supportibility from the oil wedge. A motor broken in slowly and deliberately will last much longer, but maybe be a few hp down.

In closing, the "break-in" period in modern engines has been significantly lowered, and much of the variability taken out by CNC machininig methods.