A toilet should not overflow unless the drain line is plugged, regardless of whether the tank keeps running or not. With maybe a few exceptions, all toilets have an overflow tube that provides a path for excess water coming into the toilet's tank to go into the bowl. If the bowl isn't clogged, then that water runs down the drain (think teakettle - you keep filling it, but it runs out the spout/drain). So, if the toilet frequently clogs, maybe a new toilet or some serious talk to the kids about how much toilet tissue they use is in order. take away their allowance for a few months, or their computer priviledges or TV watching time. If the tank has been damaged, or the overflow tube was replaced for some reason and is too high, then water can come out of the tank before it hits the overflow, but other than that...it should do what it is designed for - prevent an overflow.
On the old, 5+ gallon tanks, not sold for probably close to 20 years, if there was too much in the bowl that blocked the flush, there was enough water in the tank to cause it to overflow. If the fill valve kept running, then you'd have a big mess, but again, it should stop once the tank refills.