I installed a new cassette and chain. But I can't make the rear derailleur adjustment required for clean shifting. If I tighten the cable, I can't shift from the largest cog to the next-to-largest cog.. But if I loosen the cable, I can't shift from the smallest cog to the next-to-smallest cog.
I have a Shimano 105 rear derailleur and a Shimano 12/28 cogset on a 1992 aluminum Trek 1440.
You didn't mention whether the new chain is technically compatible with the cassette, but assuming it is and that there are no other compatibility issues, the main possible causes for this are limit screw misadjustment, cable friction, derailer pivot slop, and derailer hanger misalignment. The most likely limit screw problem is that your high screw is too far out.