You can put one 1600mhz and other of 2400mhz but Mixing RAM speed, however, is a slightly different matter. In theory, if you had to, you could mix, say, this Patriot model (DDR3 1333) with this Corsair model (DDR3 1600), since they have the same cas latency, timings and recommended voltage. Your motherboard would probably just automatically underclock the faster one and you wouldn’t run into any problems. Thus, it’s possible, but note that when you start mixing speeds, a chain is only as strong as its weakest link, so to speak. Your RAM will only run at the speed of the slowest DIMM, unless you wanted to overclock the others.
None of this necessarily guarantees functioning RAM, of course — you may be greeted with the Blue Screen of Death if your DIMMs just don’t like each other (or your motherboard doesn’t like one of the DIMMs). Most of the time, though, your computer will run fine if you mix different brands, sizes and speeds. If you’re just doing it to cobble together a second PC from old parts and don’t want to spend any money, that’s fine — but if you’re adding RAM to your current computer, I’d recommend getting the exact same type of RAM you already have installed. Memory is cheap enough nowadays that you’re probably better off just buying some new DIMMs and calling it a day — that’s the only way you’re going to get the best performance out of your PC.