The answer to your question is technically "Yes"
They belong to the same family/order of Characins.
The piranha (many species) would then fall under the the sub-family of the above which is Serrasalminae
Actually i've seen quite a few types of tetras that do behave in quite a piranha-like manner, i'd definately say that piranha's are essentially big nippy tetras with teeth, not just because they are in the same family but also because i've often noticed very similar behavior in them and certain types of tetra etc .
Some tetras remind me of piranha's in the way they eat too though, like quickly darting towards the peice of food to nip a small chunk out of it, darting back and then quickly turning back up to nip the food again- basically like how piranha's eat etc. A lot of tetras are also shy towards people (like piranha's), but even though they are agressive and nippy fish, they are also sociable towards each other etc.
One of the defining characteristics of Characins is that they all have toothed jaws. Almost all fish have teeth of some sort, but not all have teeth on their jaws. All Characins do.