Why does find -mtime +1 only return files older than 2 days?

The argument to -mtime is interpreted as the number of whole days in the age of the file. -mtime +n means strictly greater than, -mtime -n means strictly less than.

Note that with Bash, you can do the more intuitive:

$ find . -mmin +$((60*24))
$ find . -mmin -$((60*24))

to find files older and newer than 24 hours, respectively.

(It's also easier than typing in a fractional argument to -mtime for when you want resolution in hours or minutes.)

Well, the simple answer is, I guess, that your find implementation is following the POSIX/SuS standard, which says it must behave this way. Quoting from SUSv4/IEEE Std 1003.1, 2013 Edition, "find":

-mtime n
     The primary shall evaluate as true if the file modification time subtracted
     from the initialization time, divided by 86400 (with any remainder discarded), is n.

(Elsewhere in that document it explains that n can actually be +n, and the meaning of that as "greater than").

As to why the standard says it shall behave that way—well, I'd guess long in the past a programmer was lazy or not thinking about it, and just wrote the C code (current_time - file_time) / 86400. C integer arithmetic discards the remainder. Scripts started depending on that behavior, and thus it was standardized.

The spec'd behavior would also be portable to a hypothetical system that only stored a modification date (not time). I don't know if such a system has existed.

Fractional 24-hour periods are truncated! That means that “find -mtime +1” says to match files modified two or more days ago.

find . -mtime +0 # find files modified greater than 24 hours ago
find . -mtime 0 # find files modified between now and 1 day ago
# (i.e., in the past 24 hours only)
find . -mtime -1 # find files modified less than 1 day ago (SAME AS -mtime 0)
find . -mtime 1 # find files modified between 24 and 48 hours ago
find . -mtime +1 # find files modified more than 48 hours ago

The following may only work on GNU?

find . -mmin +5 -mmin -10 # find files modified between
# 6 and 9 minutes ago
find / -mmin -10 # modified less than 10 minutes ago