As Max Planck once remarked in a lecture,
A living and flourishing theory does not avoid its anomalies but searches them out, for the stimulus to further development comes from contradiction, not from confirmations.
The point to keep in mind is that although falsifiability, rather than verifiability, is the most important criterion in determining whether a theory is scientifically meaningful, its usefulness for the greater task of building confidence in a theory is limited. A theory is accepted not simply because it has withstood many attempts at falsification, the need for such tests notwithstanding, but because it leads to predictions that are experimentally verified. After all, the purpose of a theory is to be productive and not just to fail to be wrong.