You know, for spans and such you'd probably all be interested in the forestry's span calculator
http://www.awc.org/calculators/span/calc/timbercalcstyle.asp
Doesn't work with trusses, but it's what I used to determine my rafters and I get the sense HarryBack has been there before. He talks about dead weight, and live loads, deflection, he knows his stuff. This is how I understand how to use it. But, to warn I follow HarryBack's statement about trusses. Unless a truss is designed to have a floor or weight on the bottom you don't know what effect it will have on the entire "truss system". They can't even have holes in them if I'm correct, but again I'm not a carpenter and my sense about them is they're designed with surgical precision to handle a particular load, and if another load is brought into the equation they weren't designed around, you might screw up the balance and opposing forces designed into them and probably not pretty. So, if you don't have a truss system
Species: Spruce-Pine-Fir <most wood at lumber yards is one of these>
Size: 2x6 <in this situation>
Grade: No 2 <is usually what you get at the lumber yard, your ceiling joists may be stamped with what they are>
Member Type: Ceiling Joists <It lets you pick the most options later for loads>
Deflection Limit: I'll explain. It's marked as L/360, L/720 but what does that mean? It translates into how much sag you're willing to accept. Say for example you do NOT want your ceiling to sag more than 1/2" and your joists are 12' long. Let's see if L/240 is over 0.5". You'd take (12' length / 240) * 12 inches = 0.6" sag on a 12' span. Nope, too much. Let's try L/360. (12/360)*12 = 0.4" which, less than a half inch and the closest to our max so the one we'd pick in that situation.
Spacing (In): 16
Exterior Exposure: No & No
Live Load (psf): 30 <This is how much weight we'll be putting on it>
Dead Load (psf): 10 <This is the weight of the ceiling drywall, weight of the wood itself, and weight of the floor going on top of it>
Click calculate and you'll see we can't put 30 psf up there. The max length for 30 psf with a 10 psf dead load is 10' 3" and our distance is 12'. Changing the live load to 20 psf, is just a hair short but should work because our deflection was also a bit short of 1/2". I don't think it will apply to DavidV, if he doesn't have trusses, and puts 2x6's on top of his current 2x6's for the entire length (or as far as he can without hitting the roof) he'll make a stronger support than just a 2x6 alone.