Could any provide examples of using 'but again' please?

I'm a bit confused with the phrase's usage. I somehow tend to use it occasionally but am not sure if my use is correct. Could anyone help me on this please?

Person 1: I heard she was stupid.
Person 2: I heard she was smart, but then again you could be right...

Skippy provides a good example of "(but) then again", but there are also situations where you can use "but again". For example, if you're reiterating something that goes against what someone else is saying. Just off the top of my head:

A: Judging by the things he's said, I don't think he's very smart.
B: So what if he said some silly things in the past! I think he's pretty intelligent.
A: But again, he said that two times four is twelve. I just don't see how you can find him intelligent.

Thank you, I'm getting the idea gradually. So can I say I don't want to see him again, but then again I miss him already?

How about this:

"Some prognosticators see the current current economic meltdown and massive de-leveraging leading to a downward death spiral towards oblivion and the end of western civilization, perhaps even the End Tmes. But again, maybe they're wrong?"

I am familiar with the following:

but then again
but there again

both meaning "however"

______

but again
yet again

referring to a statement or action already mentioned