Can Initial Potential be increased through events?

Four is the ceiling, and it is per Aniimo. Once an Aniimo has taken four increases from adventure events, that route is closed for it and nothing further comes from attending. Treat those four as something you spend deliberately, on an Aniimo you actually intend to keep and build, rather than on whatever happens to be in your party when an event opens.

Attending the Cross Tea Party gives you a chance at an increase, not a guaranteed one. Showing up is the requirement, but the raise may simply not land, so several attendances can pass without moving the number. There is no way to buy the outcome.

The other constraint is timing. Special events run for a limited window and bring their own quests and temporary map zones with them, so the opportunity exists only while the event is live. If you are not attending during that window, you are not getting the increase, however much you want it. Check what is running before you plan a session around anything else.

Not settled: Which adventure events other than the Cross Tea Party carry this chance is not settled, and neither is how likely the increase is on any single attendance.