I had my shifting go all to hell when my cables quit on me. Eventually, it would get where I'd lose pretty much all tension on the cables and I couldn't get into any cog in particular. The large end of the cassette was the worst, because that's where you need the most tension on the cable. After 6500 miles on one cassette and chain, it never got that bad. I did have issues in just a couple of cogs when, after a really mucky ride, I cleaned my bike too well and had too much friction going on with the cables under the BB. The chain and cassette were fine. Lubing the cables fixed that.

4000 miles is probably near the end of the chain's life. If the chain checker is still good for sure (on the smallest measurement side), then I'd look to the cables. It could ultimately come down to being time to replace everything. Your cassette might still be fine--they can check that with the chain whip thing. Chain and cables replacement would be the cheapest way to start.