score:5
This depends on the size and popularity of your route. A small bus between two rural communities that have at most few hundred people might not even run if the driver has a bad mood that day, but an InterCity train will almost always be on schedule. In most cases you should allow yourself 5-15 minutes of delay, nearly all major trains will arrive within this timeframe.
Public transport in Budapest is generally good, even if there are delays, the system makes up for it with frequency and alternative routes.
Buses between cities (especially that do not use highways) tend to behave erratically, as in they can sometimes accumulate quite large delays over the course of their stops due to either a few problematic passengers (who don't understand that this is not the bus they were waiting for, who wish to pay with a huge banknote that the driver has no change for, tries to pay with a credit card, etc.) or many little problems (dozens of people scraping change from their pockets while others are waiting behind them, an entire school class boarding a bus, etc.).
So all in all, the generic answer is they are reasonably punctual in most cases, but there are exceptions. If you mention your exact route, I might be able to give some better estimates (if I have any experience regarding that specific area).
Upvote:1
Regarding trains, there's a joke about the Hungarian Railways (MAV): it only has five adversaries, the four seasons and the passengers. Weather related very severe delays are quite common. It can be a large tree branch falling over some tracks, killing the entire route. In the summer heat, there will be speed limits. Let's not even talk of snow. The system doesn't have a lot of redundancy built in. So, yes, normally trains will be punctual within 5-15 minutes but you need to pay close attention to up-to-date information which is typically only available in Hungary (for example, on http://www.mavcsoport.hu/ under Mavinform).