Bdtyktl

A Bar Mitzvah is a ceremony which takes place on the 13th year of a Jewish male’s life (to the day).

How could it exist that [of two Jewish boys] Jim would be born on the first day of the month and John on the last (of that same month and year), yet 13 years later John reaches the age of Bar Mitzvah before Jim?

Hint related to final answer:

https://en.wikibooks.org/wiki/Mathematics_of_the_Jewish_Calendar/Anniversaries

the giveaway is that the boys are

Jewish, meaning they follow the Jewish calendar. There is also an unique concept of Rosh Chodesh

How?

Credits to full description here: http://download.yutorah.org/2013/1053/Purim_To_Go_-_5771_Rabbi_Willig.pdf

But taking an important extract out of it :

Here too, since Adar has only 29 days, he becomes a bar mitzvah on the 1st day of Nissan.
However, the Binyan Zion (158) rules that he becomes a bar mitzvah on the first day of Rosh Chodesh Adar, which is the 30th day of Shevat, since he was born on the first day of Rosh Chodesh Adar. This novel view, which makes him a bar mitzvah over a month earlier, is based on an unproven thesis that the status of Rosh Chodesh can determine a birth day.

John was born on the last day of, say, February, while Jim was born on (for example) March 1. The exact months aren't specified, so John could be born on the last day of a previous month.

Alternatively (with the edit):

Jim isn't Jewish.

Also possible (with the edit):

Jim is a girl, and celebrates her bat mitzvah, not bar mitzvah.

Note: There was another edit, now ALL my solutions are invalid. At the time of posting they were valid, though.

Now possible with new edit:

Jim dies before he turns 13.

Might be wrong but:

It doesn't say they were born in the same year, just same month. So John was born at the end of the month one year and Jim was born at the beginning of the month a year later. Thus when 13 years from when John was born he reaches Bar Mitzvah age and not Jim

Not familiar at all with the customs, but

Maybe John was born on a leap day (29 February, or maybe an equivalent in the Hebrew calendar), and it's customary to move the Bar Mitzvah to an earlier date in case the exact date doesn't exist that year?

Because:

Jim gets cryogenically frozen and loses a month of life experience and growth. Suddenly John is older!