Hey guys! Please note that the pronunciation of "men" is a little wrong here. Please listen to the correct one here: http://da.forvo.com/search/men%C2%A8/da/ and http://da.forvo.com/word/jeg_l%C3%B8ber%2C_men_det_regner/
'But' is used to connect two clauses where the second suggests contradiction with the first, e.g.
Person 1: Are you going to ask her out again?
Person 2: Well, I like her, but I don't think we'd make a good couple.
The first clause gives a positive piece of information, i.e. that he likes her, followed by a negative piece of information, i.e. that they would not make a good couple. The order can also be the other way round, i.e. negative - positive, but they can't both be positive or negative.
In this sentence, "I am running, but it is raining", the two clauses don't follow the pattern of positive - negative. However, as several other users have pointed out, the two word conjunction even though would make sense in this sentence.
I wonder if it could be a different situation:
Person 1: Don't you usually run in the morning?
Person 2: I run, but it is raining.
meaning "I usually run, but it is raining. So I didn't run this time, but when it stops raining, I will continue my usual running. Or maybe I won't let this stop me, but it is an inconvenience. Perhaps I just look less happy or hesitate first and do it anyway.
Yes, both "I run" which is the present tense used for habitual form and "I am running" which is the present continuous form or progressive form are both translated as "Jeg løber". Perhaps I should have explained it differently. I run. It's what I do. I do it regularly. then you have the word "but" so something is different "It's raining." It does not necessarily mean that I am running right now.