There is nothing in the ENVI licensing that would cause your files not to open in ArcGIS. Do you get an error message opening the files? What version of ArcGIS are you using? What is the extension on the name of the file you are trying to open? ArcGIS expects ENVI format files to have a .dat or .img file extension and an associated .hdr with the file.

That is right - I noticed that if the filename doesn't include the ".dat" extension then it's not recognized. That appears to be what was causing my problem.