Why can't I define projection (ASCII file)?

[LeahWilson]

I have about 10 ASCII files from the same governmental source imported to my ArcMap project. Half of the files came in with a defined coordinate system and projected fine. The other half have an undefined coordinate system and I cannot figure out how to define them. I have checked permissions and I have write permissions for all. The metadata tells me the file has a datum of NAD 1983 and a Transverse Mercator projection. I have tried defining the projection as a multitude of various coordinate systems to no avail. I keep getting this box... 

What am I doing wrong?!

[MelitaKennedy]

My guess is that there's something funky in the text files--control characters, spaces in column names (leading or trailing spaces), etc. Can you add them to ArcMap using Add XY Data? If so, I would just export and/or reproject from there. If you can, you might post one of the problematic files, and one that works for people to take a look at.

Melita