It's not true in London. Trains delayed by 4 minutes count as late.
There are different times for when you can actually do anything about it - mostly if a train is delayed by an hour you can get a refund (if you chose not to travel) or compensation (if you were already travelling before the delay happened), but it's a bit complicated.
For London tube trains you can get some refunds if the train is 15 minutes late.
> Transport for London (TFL) offers refunds if a passenger's journey is delayed by more than 15 minutes. For tube passengers, this amounts to the fare for the single journey you were making, whether you have a season ticket, or have purchased a single fare. London Overground users get the same, but only when their train is delayed by more than 30 minutes
That may be true in London, for all I know, but it's definitely not true in western culture as a whole.