When does a boy become Bar Mitzva?

Many times when the halacha speaks of years it doesn't mean full years but means years as counted by Rosh Hashana. For example, for the first 3 years you can't eat the fruits of a tree. The 3 years are not complete years, for example if you plant a tree in the summer, on Tu Bishvat the first year already ends after only 6 months. The first Mishna in Rosh Hashana lists off all of the Rosh Hashanas for various things.

The Gemara in Erchin (18b) lists 5 things (קדשים, בתי ערי חומה, שדה אחוזה, עבד עברי and בן ובת)where when the Torah says years it refers to full calendar years. The Gemara says that everyone agrees that by Bar Mitzva a boy needs to be 13 years and 1 day old.

However, the Gemara uses the language מעת לעת. Rashi says that not only do you count days but you even count hours. What this means is that a boy born on כ' אייר at 3PM will only become Bar Mitzvah on כ' אייר at 3PM. Until 3PM he is still a קטן. Tosafos (Erchin 31a, Nida 44b) argues on Rashi and holds that by Bar Mitzva we don't count hours, we only count hours by קדשים and בתי ערי חומה where there is a special limud.

להלכה, the Magen Avraham או"ח סימן נ"ג ס"ק י"ג paskens like Tosafos that we don't count hours as does the Bach and the Shach (ח"מ סי' ל"ה).

Some are machmir for Rashi and have the Bar Mitzva boy redo the mitzvos after the time.

For an interesting analysis see the משנה למלך הל' אישות ב:כ"א.