See Webster 1913's definition #2.

If you are a cusp, you were born on the 21-23 day of any month, and your true astrological profile may overlap into the adjacent sign.

for example...

I was born in Virgo on the 22nd of September in the Age of Pisces. As of 2000 we are now in the Age of Aquarius, and for whatever obscure reason the crossover to Libra is now September 21. I am supposedly a Virgo but never identified. Since learning about this phenomenon I have always fancied myself a Libra.

I am not an astrology geek