Why does the water in a Powerade bottle tastes a little like Powerade after many refills?
Typically water bottles used in sports or biking are manufactured from high-density polyethylene (HDPE), which is essentially like a longer-chain version of paraffin wax. In short, it's basically a solid nonpolar hydrocarbon (with the exception of the Zeigler-Natta catalysts, which are only present in trace amounts).

Meanwhile, a quick stroll through Fenaroli's Handbook of Flavor Ingredients shows that many of the flavor compounds typically found in foods are either somewhat nonpolar, have nonpolar sections, or have (mostly unsubstituted) aromatic rings present. For example, ethyl valerate, one of the primary constituents of apple aroma, has a butyl chain, is fairly nonpolar, and is nearly insoluble in water.

However, it may be soluble in HDPE, and so it may infuse the bottle with the flavor agents so that the taste can remain for a long time.

Of course, doing a double-blind test may be your best bet for checking to see if this is a real phenomenon, or your imagination. But I've personally noticed this as well, so it could very well be real.