This is maybe a variation on an old M$ mis-info...
Came across this issue a couple of years back; changed CD - got another from M$, problem seemed to go away. Have since used the original NT CD several times without problems... (on the same and other PC's - I have several copies of NT and several PC's).
Symptoms were similar to yours... (cleaning the CDROM made no difference; and it was brand new Dell system...)
A chap on the Virtualdr.com NT forum first alerted me to the M$ mis-direct...
His advice: it's really a RAM issue... Check your RAM. It may not be properly installed...; it may be defective (NT is really 'picky' about RAM). It may not be a CD/CDROM issue - as my own experience atests...
It won't hurt to clean the CDROM unit; there are cleaning kits available that I've used with success.
The other route to suggest:
Create a smallish C: system partition (FAT16) - make it around 250Meg (even 500Meg) if you have lots of HD space...; make it bootable (sys c: off a boot-disk - ideally DOS 6.22); copy the i386 folder to the c: system partition. Install NT from there. You will need a boot-disk with CDROM support obviously; http:www/bootdisk.com is one place to get one... (but I get the feeling you probably have one already...)
c:\i386\winnt /b /x /s:c:\i386
is the command.
/b = no floppu install;
/x = make no set-up floppies;
/s: points to location of the winnt.exe (in this case c:\i386)
Install into the extended area; the boot/start-up files will go into the C: system FAT16 area. You will also have created the dual/multi-boot requirement should you wish it...
If this is a low security installation, then you could install DOS6.22 (with CDROM support) first; then install the NT as outlined above. Doing this allows easier recovery options in the future...
HTH...