That's a little overly simplistic from wherever you heard it and on face value not accurate.
You can inhale a lungfull of helium and talk funny and be fine. You might even be able to do it a couple of times in a row before you get lightheaded. And that isn't going to kill you even if you did pass out.
Generally speaking... the way your body works is, it wants oxygen BUT your body doesn't respond to lack of oxygen so you'd know it... rather your body responds to build-up of CO2. So, normally, if you're in an oxygen lacking environment you are building up CO2 in the air and increasingly breathing that and your body responds gasping for air... but it's not because of lack of oxygen, just the detected too much CO2.
Here is where something like helium comes in... Breathing helium deprives your body of the oxygen it wants BUT does not create new CO2... so if you had a setup that ensured steady flow of helium and excavated any excess CO2 that might be there in case oxygen is present... then you have a scenario where you'd pass out from lack of oxygen and eventually die and your body would never react to the lack of oxygen in a way you'd know it.
Suffice it to say, it's just a little more complicated than breathing helium from a balloon. There are threads on here discussing this in more detail if you look around.