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. This shows that the spectrum is real.
To show that all eigenvalues are distinct one applies Sturm's theorem to the sequence of characteristic polynomials.
On the other hand, if you allow all $c_k=0$, for example, you can have a Jordan cell which is not diagonalizable.