Database performance and size

I was wondering if someone could shed some light on this debate; Some people I've talked with think that the performance of a database will be
affected as the number of rows in the tables grow. Others say that performance
won't be affected at all no matter how large the database gets. Which opinion is true? I know this sounds trivial, but it's seem to be one of those questions that tests theory and real world experiences.

                                           regards,

                                           Mark