A tip for you, dear reader.
Chains and cassettes wear together. Let a chain go too far, and your cogs will wear to match it. Then, a new chain will skip on the worn cogs -- and you have to replace the cassette too. (In my experience, chainrings are a little more forgiving.)
More experienced mechanics are shouting "duh!" at me right now, but it's a basic rule that bears repeating. I learned it today when my sparkly new chain absolutely refused to hold the most-used cogs of my old cassette.
Luckily, I figured it out before I did any out-of-the-saddle efforts, or my dangly bits might have become acquainted with my stem in a soprano-singing sort of way. Ouch.
New cassette from the depths of the parts box: Check! Now we're good to go again.