We purchased a 50 year old cape cod with an unfinished upstairs. The floor joists are 6 inches. Are these sufficient to hold the weight of two bedrooms? The floor joists on the first floor are 8 inches. Why the difference?

The reason there is a difference between your floor joists is probably because your attic was never intended to be finished.

There are a few extra things to consider but out right I would say that no, 2x6 is not sufficient. You will need to do some structural changes.

What you need to figure out is the measurement between the center of each floor joist in the attic. 24 inches on center is pretty common and that usually requires using sister joists to strengthen the floor for a loaded living space. 16" oc is usually the standard for living space floors as are thicker floor joists.

Another thing to think about is the span of the joists. They can only span so far before the will snap at the bottom on up. And you may have some load bearing walls the the joists are resting on that shorten the span of those 2x6s. I would say that your best bet is to get someone qualified to take a look for you.