I was looking up some lyrics and came accross "yakışmamıştım".

I discovered this is the Past Perfect Dubative, negative, of "to pair".

I was curious: what does this translate to English as? Are these uncommon tenses? That not everyone would know?

No it is a quite usual form. It often tranlates to English with past perfect / pluperfect but often only with simple past tense. tunci writes about it beautifully here: the speaker is narrating a past action which he had witnessed.

               http://www.turkishclass.com/forumTitle_55447, post 10

Btw, I think this usage is the reason why Turks often overdo past perfect when they speak English  -  if you have noticed.

I don´t think I have noticed Turkish speakers using had or have excessively but now that you mention it, I remember during my trip, someone with extensive English used either had or have and it didn´t sound quite right.

In terms of pluperfect, I think this is ´had´ so I think yakışmamıştım would be "I had not paired". From Tunci it sounds like it could be related to a narrative tense. Of course things can´t always translate literally.

But reading the lyrics of the song, it is interesting that in Turkish, you can rhyme by picking many verbs and congujating them in the same tense.

I often use had in English to say something happened before an event.