We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the
d
d
th Veronese embedding of the projective
n
n
-space
P
n
\mathbb {P}^n
have the expected dimension, modulo a few well-known exceptions. It is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of Alexander and Hirschowitz in 1995. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e.,
d
=
3
d=3
, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension
n
n
of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of
P
79
\mathbb {P}^{79}
in
P
88559
\mathbb {P}^{88559}
.