Am I able to use my lenses for my 35mm camera, on a Digital SLR Camera of the same brand?

It actully depends on the brand, and body style of your camera.

For example:

Nikon SLR lenses are universally compatible with Nikon digital SLR camera (for couplings made after 1975ish).

You will loose some functionality when you mix and match AF and MF lenses.

Canon Digital Camera's, however, are only compatibles with auto focus lenses from the same body series.

If you have EOS lenses they will work with all Canon Digital SLR models.

If you have A-1 body style lenses (which was another very popular 35mm series) they will not work with the current line of Digital SLRs.

If you can tell me what 35MM you've got, I can give a list of compatible digital SLRs.

Yes, I have done just that, and it works fine.

With the Nikon F lenses system, i think you can
however, in some cases, you won't have all feartures on using in a DSLR body.
If u are using a AF lens of Nikon with your previous SLRs, then u can stick it on the DSLR as well, and you can use it easily, everything works beautifully.
AI-S lenses of Nikon is manual focus and can't use the autofocus features on a DSLR. No metering at all. U can stick in your camera, change the aperture on the lens, set the speed manually, and shot! Everything is fine, but manually

That's about Nikon.