In Nazir 2:10, the Mishna states:
:אֵין תִּגְלַחַת פָּחוֹת מִשְּׁלשִׁים יוֹם
There can be no shaving for a period of less than thirty days.
Meaning, if you were in the middle of one nazirship when you started a second one, and you had less than thirty days left of the first nazirship, you would still need to finish the first nazirship thirty days after the end of the second one. This is because you cannot end two nazirships within thirty days of each other.
What is the source/reason for the thirty-day minimum between ending nazirships? A nazirship must be at least thirty days, but surely the days of the first nazirship before the second one would count for that!