Can one mix bitcoins by oneself? Do mixers rely on other users, or can one mix his or her own bitcoins alone, such as by sending them to one's own addresses?
No, you do need a mixer. Even if you were to generate a million random transactions with your wallet, 100% of the value in the unspent output from the mixing could be linked to the original wallet through simple breadth-first search.

Transactions are value-preserving (except for the coinbase), have one or more inputs, and have one or more outputs. So you can trivially construct a transaction graph, which is a directed graph with transactions as nodes, inputs as head ends of edges, and outputs as tail ends. You also know the graph is acyclic, like a family tree: a transaction A can't both be the ancestor and descendant of B. With that in mind, if you start at transaction B, which pays your bank and is traceable to you, and go backwards one level at a time (follow edges in the opposite direction of the arrow), all paths from B to their respective coinbases go through A. 

This is strong evidence that A and B belong to the same person, just like ending up with a briefcase full of marked bills is strong evidence that you got those bills all at once and just shuffled them within the briefcase, as opposed to getting one marked bill as change from every cash purchase you made for three years, out of pure bad luck.

This is the same as saying you have a bottle of vodka that you want to dilute. Pouring it into 100 glasses, then pouring it back into another bottle, doesn't hide the fact that it's vodka.

Mixers work because the inputs they receive are from different sources. If you're the only one pouring vodka into the punch bowl, and everyone else is pouring fruit juice, then when you take back the same volume of liquid you put in, it has less vodka, maybe not enough to even be detectable.

Of course, if an adversary wants to know you used a mixer, it's fairly trivial: they deliberately send coins they control directly to the mixer, and see if any of your addresses has ever been tainted by these coins. The analogy is someone pouring blue ink into the punch bowl, then the cops arresting anyone under 21 exiting the party with blue lips. They still can't know whether you put vodka or fruit juice into the bowl, but using a mixer may itself be considered money laundering in your jurisdiction.

Another issue is the other users of the mixer. Those who use a mixer at all likely have something to hide, and some proportion of the coins you get may come from shady sources. Of course no one would get rid of perfectly good fruit juice, but they want to get rid of their extra cough syrup, rat poison, lead paint, and drain cleaning acid. Sure, you got rid of your bootleg vodka, but what you got back isn't exactly safe for kids.