If you mean version 6 which as I recall was called ColdFusionMX, then I think you are going to find it very hard or will simply not work. CF8 and CF9 work on Windows 7. The issue in running CFMX on Windows 7 may differ according to what web server your using be it IIS, Apache or “built in CF”.
CF was CFMX until CFMX7: CF8 was the first without it. But, yes, you're right: knowing exactly what the OP is talking about would be helpful here.
I haven't had any problem installing CF8 or CF9 on Win7 (just the Starter version of Win7 for me... it's a netbook). Then again, I had no problems installing or 7, 8 or 9 ( simultaneously) on Win Vista (Home Premium) either. So the difficulties you encountered installing on Vista were not down to CF, so if it's an idiosyncracy of how you're doing the install, or the environment when installing (eg: not being logged in as an administrator), then you're quite possibly going to have hassles on Win7 too. Still: the easiest way isn't to guess, or to ask other people: it's to try it.
If the question is simply "is it supported", then there's a web page for that:
(note that clearly the "supported" matrix is a subset of the "will run on" matrix, given I've had no probs on the unsupported versions I list above)
Versions of CF earlier than CF8 are being end-of-lifed, and are not actively supported by Adobe. This is not to say they won't work, they're just not supported. One should not be running a version of CF older than CF8. Not least because earlier versions are pretty bloody slow and less stable compared to CF8.