Yes, you can sue someone for the breach of GDPR. This data protection law gives you the right to claim compensation against the suffered damage. The data breach can cause both, material (loss of money) and non-material damage (such as stress or anxiety). So, you may make a GDPR breach claim for either of these losses.
However, it is not necessary to make a court claim, as the offender may simply agree to compensate you. If, because of any reasons, he does not agree, you may go to the court for making the claim. The trial in the court would decide whether or not that person or organisation would have to pay you.