The error basically means that the server is too busy to handle the request at the particular time.
The reason for this can of course vary, one reason that free hosting is more prone to then paid hosting (regardless of host involved) is that there's people that with intent abuse the service given by basically hijacking the server resources, if there's enough abusers at the same time this would obviously cause an issue as there would be no resources left for the normal page views.
Not all abuse is intentional of course, it can be caused by badly coded scripts that fail to release the resources properly as well but the end result would be the same as intentional abuse, there's no free resources available at that time.
The problem with solving these kinds of problems is finding the abusers, they may not be apparant unless you manage to catch them in the act, and to be able to do that it means someone would constantly have to monitor what happens every second. We have monitors that alerts when something happens of course, but there may still be a few minutes before abusers are found and shut down causing temporary issues.
As for if it will happen on paid hosting, that would be very unlikley, first of all the intentional abuse is almost completely abscent, there would be money lost if they are caught as no refunds would be given. As for the unintentional abuse, yes that could in theory cause a problem but is very unlikley to do so as there are more resources available per user on paid hosting, there's far less users on each server.
Does the problem always occur at the same time? if so could you let us know what time they occur, as well as what timezone you are in so we could try to pinpoint what happens on the server at that given time by checking logs.