Part 15 sets limits on unlicensed RF emissions. The limits vary by frequency, the type of device, the nature of the emissions, and so on. It's a pretty huge document and you need a lawyer and an EMC engineer to approach it. However, you can get the intent of it from this blurb that is printed on just about every piece of consumer electronics sold in the US:
This device complies with part 15 of the FCC rules. Operation is subject to the following two conditions: (1) this device may not cause harmful interference, and (2) this device must accept any interference received, including interference that may cause undesired operation.
It never hurts to ID. However, if you are testing an oscillator that isn't even connected to an antenna or an amplifier, or if you are using a dummy load and modest power (less than 10 watts, say), your effective radiated power is going to be less than a microwatt.
If you are in doubt, listen with another receiver. Can you hear your device under test above your switching power supplies, the power lines, touchlamps, and ambient atmospheric noise? If not, then it's unlikely you are "causing harmful interference". If you don't have a second receiver, it's not terribly difficult to construct a field strength meter.
This doesn't per se mean that you are complying with part 15 (that could require filing with the FCC, passing a compliance test, ...) but at some point, a little common sense is sufficient. You aren't mass producing a device that will be used by millions of people for years and years. You are performing a temporary test as a licensed amateur radio operator. Take reasonable steps to avoid causing interference and you will be fine.