While watching the John Adams miniseries, I noticed that he did things that would be unthinkable to Presidents today, such as stumbling around a dirty slum with no security protection. Wouldn't somebody notice and say "Hey, that's the President! Let's mug him!" or something?
I believe that your question cannot be answered apart from a cases by case basis, it would depend on the celebrities in question how popular they were and how much they would accept to step down in order to not be recognized. Here are two examples where celebrities where recognized even before photography.
One such episode six years before Adams becomes President of the USA occurred in France during the so called Flight to Varennes. In June 1791, Louis XVI trying to escape from Paris was recognized by a postmaster Jean-Baptiste Drouet in Sainte-Menehould and later arrested in Varennes. According to a political review of the time he testified that the man in the coach had a figure similar to the one on some bill. Obviously some mistakes where made in the inception of the flight: the carriage was humongous and too slow and made the soldiers stationed on the road to ensure the safety of the king suspicious in the eye of the population. We all know how it ended for the French king…
Another earlier one would be the Great Embassy of Peter the Great. The young tsar would travel in Europe incognito disguised as Peter Mikhailov. He was considered simply as a member of the embassy, even if the sovereigns he visited found a way to have a private meeting with him. It seems to have been mainly an attempt to ditch protocol and lengthy ceremonies in order for him to have more freedom. But he was too often recognized. One such exemple was his visit of Zaandam where he worked as a carpenter. Arriving Sunday, August 18, 1697 he encountered a blacksmith that had worked with him in Russia who helped them settle, he started working on Monday in the shipyard. But by Tuesday everyone in town knew that there was a person of great importance in Zaandam, and the same day he was obligated to reveal his identity to the burgomaster after an encounter with some mean boys who chased him away by throwing stones at him. And Wednesday his identity was out in the open when someone read publicly a description of the tsar and the blacksmith's wife harassed by questions finally let it slips. Many didn't believe it but it was again confirmed by merchants who had seen the tsar in Arkangel. The following day many people came to see him like a curious beast and he could not work anymore so he departed on Sunday.
Jean-Clément Martin, Nouvelle histoire de la Révolution française
Robert K. Mazie, Peter the Great, his life and world