The probability $P(f)$ that the spacing $s$ of two eigenvalues of a random real symmetric matrix is smaller than the average spacing $\bar{s}$ by a fraction $f$ is given by $$P(f)=1-e^{-\pi f^2/4},$$ according to the Wigner surmise (valid for a large-dimensional matrix under very general conditions).

So dependent on the precision of the calculation, this should tell you with what probability you will not be able to distinguish to nearly identical eigenvalues.

The probability $P(f)$ that the spacing $s$ of two eigenvalues of a random real symmetric matrix is smaller than the average spacing $\bar{s}$ by a fraction $f$ is given by $$P(f)=1-e^{-\pi f^2/4},$$ according to the Wigner surmise (valid for a large-dimensional matrix under very general conditions).

So dependent on the precision of the calculation, this will tell you with what probability you will not be able to distinguish two nearly identical eigenvalues.