It existed but was not common. The Encyclopaedia of Islam entry on the question is quite succinct:

The name "Muhammad" is reported to have occurred previously among the Arabs (e.g. Ibn Durayd, ed. Wiistenfeld, 6 f.; Ibn Sac d, i/1, 111 f.) and therefore need not be regarded as an epithet adopted later in life by the Prophet. It should be noted, however, that the brief section on such persons given by Ibn Sac d has the heading, "Account of those who were named Muhammad in the days of the djahiliyya [q.v.] in the hope of being called to prophethood which had been predicted", which indicates the tendentious nature of some of these accounts. The fact that the sources say frequently that in his youth Muhammad was called Amin, a common Arab name meaning "faithful, trustworthy", suggests the possibility that this could have been his given name, a masculine form from the same root as his mother's name, Amina. The name Muhammada for women occurs several times in the Syrian Book of the Himyarites.

For some context, the reason why it might be an epithet is that etymologically it means sort of "one who is praised", which seems awfully convenient for a prophet, hence the reference to: " those who were named Muhammad in the days of the djahiliyya [NB meaning "pre-Islamic times] in the hope of being called to prophethood which had been predicted.

The Kingdom of Himyar was a large pre-Islamic kingdom in Yemen

As another question, why is it permitted for people to name their children 'Mohammed', but images of him are haram? I apologize for my ignorance.