OK, strictly speaking, the problem has more to do with the Earth and the sun not being able to get their shit together and create a year that's composed of a nice round number of days: 365.25 doesn't divide evenly by anything, no matter what the pope says.
"God's working on changing what integers are, right now."
Still, that hasn't stopped people from trying. And 365.25 is kind of close to 364, which is 28 times 13. That means we could have nice, even 28-day months, except we'll have 13 of them, which is basically the unevenest number there is. And we'd still have a spare day each year, which we'd have to do something with. It probably couldn't be a regular day. These guys who invented a 13-month calendar want to call it the "day out of time." The guys who invented this 13-month calendar call the spare day "Year Day," which sounds less dumb until you think about it, at which point it sounds much, much dumber. And it wasn't even just kooks promoting these alternative calendars. After World War II, the United Nations was seriously considering a "World Calendar" that featured 12 months (phew) but still had those insane "blank days," which are so confusing.
None of these proposals have gone anywhere, although unlike some of the other items on this list, it's not just because people were lazy. No, it turns out that the idea of a blank day is really hard for most religions to wrap their heads around. If you're supposed to dedicate every Sunday to worshiping God and you spend the day after Saturday drinking beer in a park because that's the blank day for the year, is God going to be cool with your accounting?