  We study $p$-adic families of eigenforms for which the $p$-th Hecke
eigenvalue $a_p$ has constant $p$-adic valuation ("constant slope families").
We prove two separate upper bounds for the size of such families. The first is
in terms of the logarithmic derivative of $a_p$ while the second depends only
on the slope of the family. We also investigate the numerical relationship
between our results and the former Gouv\^ea--Mazur conjecture.
