As they say in Spain, "poner una pica a Flandes." It would be pretty damn hard. Think about this in two ways. You have to gather the resources to found a colony in the Americas, and then you have to get royal permission to do so. And not only that, you have to maintain the colony and royal blessing. Otherwise, you risk the royals licensing somebody else -- likely with more men and weapons than you -- to come and discipline you.

You can read posts related to Columbus' own experience. He initially negotiated an agreement with the royals that was very much in his favor. And he obtained investors to pay him money for an expedition. Over time, as his incompetence and abuse became more clear, we was dispossessed and brought back to Spain in custody. All the while there were rebellions by his colonists that were pacified by administrators empowered by the royals.

On his second voyage, there were other speculators attempting the same, but within a few years they were brought back under royal control.