dimoj's "attempt" is really the only way to accomplish that.
If joining them the way you are trying, you would, theoretically, end up with a path with 3 end points which doesn't happen in Illustrator.
dimoj suggests you make sure your paths are the desired thickness and use the "Expand..." command (which in Illustrator 10 is just under the Object menu). Not sure of previous versions, but after selecting expand in AI10, make sure the stroke checkbox is selected.
What you end up with is to really skinny boxes that still look like the original path. You then use the Pathfinder tools to Merge them together so you know have what you were looking for.
The only other way to accomplish it would be to split the one path in half where you want the intersecting path to "join," dupe that path and, in essence make 2 upside down L's by joining the perpendicular paths together. (Boy, this hard to describe) Then just overlap the 2 paths and, Viola!, it looks like you have a three-end path with the ability to change stroke width, etc.
All I have to say is "good luck"
bob