The spectral theorem says that if $A$ is a self-adjoint operator on a Hilbert space $H$ with compact inverse, then the eigenvectors of $A$ form a complete orthonormal basis of $H$. Furthermore each eigenvalue is real and $|\lambda_n|\to\infty$ as $n\to\infty$.
Now let us slightly change the conditions of the theorem. Let $A$ be a self-adjoint operator with increasing sequence of eigenvalues $0<\lambda_1<\lambda_2<...$ Is it true that the eigenvectors of $A$ form a complete orthonormal basis of $H$?
Let $H=\ell^2(\mathbb N)\oplus L^2[0,1]$. Write $\{e_n\}$ for the canonical basis of $\ell^2(\mathbb N)$. Now define $A$ by $$ Ae_n=\lambda_ne_n, $$ and on the second component let $(Af)(t)=tf(t). $ Now the eigenvectors of $A$ are $\{e_n\}$, which do not span $H$; in fact, their orthogonal complement is infinite-dimensional. You can tweak the second component to make the spectrum of $A$ almost anything you want.