If you'd like some, you know, physics from your physics blogs, here you go: Andrew Jaffe points out new results on neutrino oscillations from the MINOS group, providing new limits on the differences between the masses of different neutrino flavors. You can also read the Fermilab press release, which as a bonus contains some wonderful examples of stilted "quotes" constructed by cutting and pasting text from emails.
I've recently become sort of tangentially (very tangentially) involved in efforts to detect both neutrinos and dark matter, so I'm a lot more interested in these sorts of stories than I used to be. My knowledge of the details of how this stuff works is still pretty sketchy (in particular, I'm not sure whether they can use the mass difference they measure to determine an actual mass), so any explanation I could give would be almost indistinguishable from just telling you to go read Andrew's post. So, go read Andrew's post.
(in particular, I'm not sure whether they can use the mass difference they measure to determine an actual mass)
Since masses are positive, the lower bound on A - B gives a lower bound on A...
Chad-
Alas, the answer is that they can't measure individual masses. I've added some more info to my post...
-Andrew
There are some interesting cosmological bounds on the sum of the masses, however. I think these are in one of the new WMAP papers or another. I'm not sure how much they depend on the modelling of structure formation, however.