All real tridiagonal matrices with $b_kc_k>0$,  are diagonalizable, and their spectra are real and simple.

See, for example, Gantmakher and Krein, Oscillation matrices and kernels..., AMS 2002.

Sketch of the proof. Expanding the determinant $|A-\lambda I|$ write a recurrent formula for characteristic polynomials of truncated matrices. It is seen from this formula that the eigenvalues depend only on $a_k$ and the products $c_kb_k$.
Therefore the symmetric matrix with $c_k^\prime=b_k^\prime=\sqrt{c_kb_k}$ has the same spectrum.

On the other hand, if you allow all $c_k=0$, for example, you can have a Jordan cell which is not diagonalizable.