About the only glaring weakness I see is the way they handle recoil. That soft aluminum recoil lug isn't too smart. The small caliber guns don't have much of an issue with it, but the hard kickers will shear the lug halfway through in no time, allowing the action screws to bear on the inside of their holes and start absorbing the recoil. At that point the accuracy goes to pot.
The older Tikka actions had a much more intelligent way to deal with recoil, and don't fail like this. But the newer one is way cheaper to make, so that got the nod. Otherwise, they are a great design. The bolt is slick, easy to dissasemble, good tunable trigger, great quality barrels, etc. Now that EGW is making a picatinny rail for them it's easy to mount a scope.
Stocks are an issue, most of the T3s I see have the plastic stock - not much that can be done about that. The wood one can be bedded, but you are still left with that aluminum recoil lug. Manners is now making stock to fit, I don't think much of the shape of his stocks but nobody can argue about the build quality - they are top notch.
That's about it - nothing is perfect, but T3s are way better than a great majority of the designs out there.