Question: I was on about 87 kg to 88 kg. With my height of 1,94 m, it was pretty damn good in terms of my BMI :) I nearly hit exactly the middle of the target value of a mid-twenty male. 
However, I started to train for a run a few weeks ago. I'm running nearly every second day (8 to 11 km) and do a few push-ups (20) and sit-ups (30) twice per day. 

I noticed that I've gained about 3 kg! Okay, I just would've thought "Yep, have to be the additional muscles" but it concerns me, that my BMI is out of range. Is the BMI reliable when it comes to added muscle mass? Should I try to stay in the target range regardless of the training?

Answer: 
BMI is just an average target, being healthy and fit varies per person. I used to be really underweight according to BMI measurements while feeling fine and working out almost daily (cycling, running) and eating properly.

I've started doing weight training about 2 years ago and gained about 20kg (44lbs) in muscle, and according to BMI I am now your average joe, even thought I'm very muscular, and able to do things that most people in my gym cannot.

What I'm trying to point out is that BMI isn't a 100% accurate measurement. Gaining a couple of pounds/kg is a good thing, don't let it put you off. You are getting healthier from doing a mix of cardio and weight training, that's for damn sure.
The respondanse have been correct, B.M.I is a flawed way of measuring fat to bone/muscle. Hip to waist measurements provide far a better idea of how much fat you carry. More sophisticated techniques give accurate measurements but are costly.
What to do??? Not always easy to do, but simple to quantify, use more calories in exercise than consumed by mouth will ALWAYS do the job. A more simple take home message " Don't eat it if it takes too much effort to burn off"
David W Cornwall