It seems that towards the end of ROTS, Yoda comes to grips with the fact that he and the other prominent Jedi masters have somehow failed to change with the times, or something like that. I don't remember the exact quotes, but the general idea is that he has a lot of regret that he and the Jedi order didn't handle things differently.

Like what, exactly, did they handle poorly? What could they have done differently that would have made them more successful, and what, exactly was the problem to begin with?

In AOTC, they make hint that "our ability to use the force has diminished". What the fuck does that mean? They can't shoot lightning very well? They can't see the future? They can't move shit around with their minds as well? Obviously, the dark side clouded shit up, but is that their fault?

Will Yoda make reference to any specific situation that he or the Jedi could have handled differently in ROTS, or we be left to figure this out for ourselves?

yoda had trained jedi to be like those who had trained him almost 900 years before. they were exactly the same. the sith on the other hand spent a millenium learning, changing, and adapting to the new world. jedi were more concerned with the "code" and "rules" than the living force, which qui-gon was a big supporter of. that's why they kept him off the council.

Also, it seems that Yoda just feels miserable after all hell breaks loose and blames himself. By his "failure," he could mean that he feels like he's failed, since most of the Jedi have been wacked.