Is an enlarged thyroid sometimes a genetic problem?

It can: Familial multiple nodular goiter and hashimoto's disease both of which have an underlying genetic cause and can leadvto thyroid enlargement/goiter.

Rarely: There are rarely pure genetic causes for goiter, like in patients with pendred syndrome and other even more rare conditions. However, most of the time, goiter is the result of a disease process within the thyroid. Genetics probably plays a role in one's susceptibility to thyroid disease as it's common to see patients with thyroid disease who also have other family members with thyroid disease.