what's your idea about "but although"

once my student used the term but although, I corrected her and asked her to use 'although' instead. He insisted he had seen this word before.

I looked up and found we actually have this word. Is it common? Can we use these two words together? I found some examples from new york times and Guardian. Probably in Ame English?

Guardian: But although I understand their arguments, I disagree.

But although the city was outraged, the murder went unsolved.

Solution 1:

But and although are entirely separate here, with different discourse functions: there is nothing wrong with using both of them.

But is connecting the sentence to the existing discourse, and expressing that this sentence is in some way differing or countering what has gone before.

Although is specifying some concession in the following sentence: this concession may or may not have been introduced to the discourse already.

But although I understand their arguments, I disagree.

and

Although I understand their arguments, I disagree.

have the same denotative meaning, but their role in the discourse is different: in the first, the speaker is giving notice that the sentence is somehow countering what has gone before, thye usual function of initial but.