Yep, sounds just like a worn chain/cassette with a new component fitted. If you change either of the chain or cassette (and sometimes the chainrings at the front) they need to all be changed because they wear together (the spaces between each link/tooth stretch out with the wear). After the drivetrain has been used for a length of time (~3000kms) changing just one part (the chain or the cassette) will lead to mis-matching and the chain will skip. Often it's jst on one or two cogs - the most heavily used, and the smaller cogs wear quicker.
Usually you can use the same chain and cluster for a long time without problems. I've seen a friend ride the same drivetrain until he had broken several teeth off on the cluster

. But if you have been tending to ride a lot of the time in a few cogs, then shifting to a less used cog (like the top gear) will show up the worn chain.
An easy trick to judge wear is to lift the chain at the chainring. A new chain will not lift off. If you can see daylight under the chain it has some wear, lift it to 1/2 the height of a tooth the chain needs replacing, lift it to the full height of the tooth then the chain and cassette will be clapped out.
This is not very scientific. You can do it more accurately by measuring 12" of chain - this will be 12 full links on a new chain as the chain has a 1" pitch. If 12 links measure >12 1/16" the chain is worn and needs to be replaced, but sprockets should be OK. If it's 12 1/8" the chain and sprockets may be too worn for a new chain to mesh correctly and a new cassette will be needed. >12 1/8" you'll need a new cassette for sure, and maybe new chainrings
Sheldon Brown says it all. His site is a golden bible for bike maintenance.
Riding bikes in traffic - what seems dangerous is usually safe; what seems safe is often more dangerous.