Yesterday I re-stumbled upon an old grammar worksheet from David Foster Wallace in which he had his students try to correct sentences that largely didn’t need correcting. I’d first been introduced to it way back in 2009, at which point I complained about its infelicitous “correction” of a split infinitive. That bit of baloney was so egregious that it made me overlook another silly claim, one nearly as common as the split infinitive rule, and nearly as mistaken as well.
The claim: that each other is to be used exclusively with two objects, and one another exclusively with three or more. This is a pretty widespread claim. Perhaps it speaks to some unconscious desire by English speakers to exhibit a form of the dual/plural distinction, since some people also insist that between and among are to be used for two and more things, respectively. But just as the between and among distinction is a bunch of made-up hooey, so too is the each other and one another distinction.
The claim has a pretty long history; the MWDEU (Merriam-Webster’s Dictionary of English Usage) tracks it back to George N. Ussher in 1785. He stuck with that idea, including it in a grammar booklet in 1803 as well. Some subsequent commentators accepted this rule, others rejected it. Still others proposed alternate criteria for differentiating the phrases, such as Thomas Marsh’s 1862 idea of “the former [each other] applying to a limited, the latter [one another], to an unlimited number”. Only Ussher’s proclamation has stuck around.
Of course, none of these proposed separations are valid, neither in current English nor in that of any other time. Examples of famous and well-regarded writers using one phrase where supposedly only the other could go, even in formal writing, are plentiful. The MWDEU lists Samuel Johnson, Noah Webster, and Bishop Lowth among others; the OED offers Shakespeare and Caxton as well. To hammer home the point, I’ve found some examples of both phrases being used in the same sentence to refer to the same set of things:
“The Genoese and Piedmontese, therefore, although both Italians, and living within a few miles of one another, detest each other as cordially as the Spaniards and the French.” 
“[...] the two aged actors upon this great theatre of philosophy and frivolity embraced each other by hugging one another in their arms, and kissing each other’s cheeks; and then the tumult subsided.” 
“[...] the parts of the coils nearest each other tend to neutralize one another [...]” 
These aren’t rare instances, either; more examples from any time period you’re interested in can be found by searching for “each other * one another” in Google Books. We can even do one better by finding a sentence in which the two phrases can directly alternate. Thanks to the many competing English translations of the Bible, we can find different ways of saying the same thing. Here’s Ephesians 4:32 in two translations:
(1a) Be kind and compassionate to one another, forgiving each other, just as in Christ God forgave you. [NIV]
(1b) Instead, be kind to each other, tenderhearted, forgiving one another, just as God through Christ has forgiven you. [NLT]
I’ll confess I got a bit giddy when I found that. They’re both equally well-formed usages to me, they show that the two phrases are interchangeable, and it’s another chance to say “God said it, I believe it, that settles it” in the linguistic arena.*
There are likely some minor differences in usage between each other and one another, of course. The only one I can say confidently is that each other is more common; Google N-grams has it appearing around twice as often, and COHA has it almost four times as often. Perhaps influenced by its relative rarity, I find one another to be stiffer than each other, but I don’t know if this is a generally-held position.
There’s one thing I’d like to know, but not enough to actually perform the analysis, and that is whether each other is indeed preferentially deployed in situations with only two objects (and vice versa with one another). I’ve no data pointing either way, nor an impression from other people’s usage if this is actually the case. But at this point, it’s nothing to worry about; at strongest, we’re talking about a preference, not a rule. If you want to maintain this distinction in your own English, I’m not going to say you can’t. But don’t get confused and think others ought to obey your whims.
Summary: Despite claims to the contrary, each other and one another are both acceptable whether you’re talking about two or more than two objects. English usage never observed the supposed rule, and great writers broke it often. In fact, the two forms can alternate with one another.
*: Just to be clear, I do not actually believe that examples pulled from religious texts should hold any special place in informing our linguistic judgments.