It says in Griffith's chapter 2.1, that:
$$\tag{2.14} \Psi(x,t)~=~\sum_{n=1}^{\infty}c_n\,\psi_n(x) e^{(-iE_n t/\hbar)}$$ It so happens that every solution to the (time-dependent) Schrodinger equation can be written in this form [...].
By "solution" does he mean solutions on separable form
$\tag{2.1} \Psi(x,t)~=~\psi(x)f(t),$
which he stated to begin with?
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No, by a solution he means any solution, i.e. a function $\Psi(x,t)$ that satisfies Schrödinger's equation and that will typically not be writable in the $a(x)b(t)$ separated form. The claim that any solution may be written in the summation form you quoted is the claim that the solutions in the separated form are "sufficient" because the most general solution may be written as their superposition.
There isn't really any ambiguity in the text.
That wave function is a series
$$\Psi(x,t)=\sum_{n}c_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar} $$
every term (of index, say $j$) looks like a separable solution
$$\Psi_{j}(x,t)=\psi_{j}(x)e^{-iE_{j}t/\hbar}=\psi(x)f(t)$$
and because the equation is linear, if $\Psi_{j}$ is a solution of the time dependent Schrödinger equation, then
$$\Psi (x,t)=\sum_{n}c_{n}\Psi_{n}(x,t) $$
is the most general solution.
