~을테지만, ~을테니까 's "테" comes from 터 .
터 means place, foundation, ground, site. So ~~~을 테지만 is ~을 터이지만, (literally, it's a place/situation where.... but) and ~~~을 테니까 means ~~~을 터이니까 (literally, it's a place/situation where.... so).
So, like Jeff said, that has the meaning of "naturally it would be + but/because" .
Different ways to say the same thing would be:
만날테지만 = 만나겠지만, 만날 거지만
못 할테지만 = 못 하겠지만, 못 할 거지만