Octopi is an example of a hypercorrection.
Some people know that English often uses plurals according to the rules of the language a word came from. (It's not strange that this happens, since such words after all aren't English in origin, so the difference is between borrowing a word and then pluralising it, or borrowing the plural word.)
These people also know that with such words, use of the English plural form is sometimes considered incorrect. They further know that, justly or unjustly, where there is a choice, the use of the non-English form is sometimes considered cleverer, because it requires further knowledge.
Virii is another common example, which may have started as a joke. Indeed, several such cases did; penii being a more more obvious example of it being a joke rather than an earnest mistake.
Opus is an interesting case, because opuses and opera are both found, following English and Latin rules respectively, but also opi, following Latin rules incorrectly; so we have both an English, a Latin, and a hypercorrect Latin form.
A hypercorrection, or any other mistake, can eventually become part of a language if followed by enough people. Since a language exists in a medium of loose consensus, then even if we don't like this, it'll be the case. An example that's dying out again (and isn't about plurals) is that for a long time foetus was the more common spelling than fetus, though the latter became the more common spelling in America some time ago, and elsewhere quite recently. Yet foetus is a hypercorrection of what in Latin was originally fetus. Indeed, I'll still spell it foetus even though I know this because it's the spelling I'm used to.
Now, when we come to dictionaries, it depends on the purpose of the dictionary in question.
A dictionary like the OED would be remiss in not mentioning octopi, because octopi is used considerably, and it's the job of the OED to record this.
With other dictionaries, there's a good argument to be made for not including it. Arguments for including it though would be:
Someone who came across octopi and looked it up, should be given an answer (perhaps with a note saying "incorrect" or "proscribed", perhaps not).
The usage panel may have decided that it's so heavily used as to count as a plural that came from an error, rather than an error.
In fact, even my google chrome and firefox spell checker tells me radiuses is incorrect.
This is a case that certainly shouldn't allow octopi; the only reason for a spell-check dictionary is to spell something correctly. Even if we were to edge toward the idea that octopi had been fully absorbed into the language as a new plural, while octopuses remains common and easy to guess at if you're confused by "why did it say that wasn't the word I wanted?" then it serves no real purpose at all.
If anything, spell-check dictionaries are too forgiving for their purposes; if one mistypes and accidentally produces an obscure word it's better to have it flagged even if there's no doubt it is indeed a word; you can add it yourself if you've cause to use it often.