This paper studies the critical behavior of the 3d classical O(N) model with a boundary. Recently, one of us established that upon treating N as a continuous variable, there exists a critical value N_c > 2 such that for 2 ≤ N < N_c the model exhibits a new extraordinary-log boundary universality class, if the symmetry preserving interactions on the boundary are enhanced. N_c is determined by a ratio of universal amplitudes in the normal universality class, where instead a symmetry breaking field is applied on the boundary. We study the normal universality class using the numerical conformal bootstrap. We find truncated solutions to the crossing equation that indicate N_c ≈ 5 . Additionally, we use semi-definite programming to place rigorous bounds on the boundary CFT data of interest to conclude that N_c > 3, under a certain positivity assumption which we check in various perturbative limits.