Dic: but again, then again, but then, there again

• It's easier to take a taxi. But then again you can't always get one.
• William didn't succeed first time, but then very few people do.
• Elaine's father might lend them the money, but then again he might not.
• She says she's thirty-five. But then again she might be lying.
• I need a vacation. Then again, so do my coworkers.
• On the other hand, she is too much ambitious for her own good.
• Then again, she might not go.
• I like to travel but, then again, I'm very fond of my home.

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△ You might use again in expressions such as but again, then again, and there again when you want to introduce a remark which contrasts with or weakens something that you have just said.

△ but then (again): used to say that although something is true, something else is also true which makes the first thing seem less important.

△ then again, there again: (spoken) used to introduce an idea or fact that is different from  something you have just said, or makes it seem less likely to be true.

△ then again: (idiom) From another standpoint; on the other hand.

△ then again: (contastive) from another point of view. = on the other hand, but then, 

△ then again: used when you had a new thought that is different or opposite to what you have just said.

△ then again: however, but, yet, on the other hand.

.... COBUILD, LDOCE, American Heritage, WordNet, CALD, Vicon English Dictionary