In my physics lessons, my teachers have always been keen to tell my class that Jupiter is considered a 'failed star' by scientists. Is this true?

In my own effort I wondered if maybe this could just be being regurgitated from an outdated physics syllabus that still considers the Solar System to have nine planets. From that thought onward, through my research on the Internet, I haven't found people referring to Jupiter as such and people always call it a planet rather than a brown dwarf. 

Furthermore, it's my understanding that brown dwarfs possess more mass than Jupiter suggesting to me that Jupiter possesses too little mass for fusion to even be plausible.

So am I correct in thinking that Jupiter is 'only' a planet, or are my physics teachers correct in saying it is a failed star (and if so, why)? 

The answer kind of depends on how old you are. At a very introductory level, say, maybe middle school or younger, it's "okay" to refer to Jupiter as a failed star to get the idea across that a gas giant planet is sort of similar to a star in composition. But around middle school and above (where "middle school" refers to around 6-8 grade, or age ~12-14), I think you can get into enough detail in science class where this is fairly inaccurate.

If you ignore that the solar system is dominated by the Sun and just focus on mass, Jupiter is roughly 80x lighter than the lightest star that undergoes fusion. So it would need to have accumulated 80 times what it already has in order to be a "real star." No Solar System formation model indicates this was remotely possible, which is why I personally don't like to think of it as a "failed star."

Below 80 MJ (where MJ is short for "Jupiter masses"), objects are considered to be brown dwarf stars -- the "real" "failed stars."  Brown dwarfs do not have enough mass to fuse hydrogen into helium and produce energy that way, but they do still produce their own heat and glow in the infrared because of that. Their heat is generated by gravitational contraction.

And Jupiter also produces heat through both gravitational contraction and differentiation (heavy elements sinking, light elements rising).

Astronomers are not very good at drawing boundaries these days, mostly because when these terms were created, we didn't know of a continuum of objects. There were gas giant planets, like Jupiter and Saturn, and there were brown dwarf stars, and there were full-fledged stars. The line between brown dwarf and gas giant - to my knowledge - has not been drawn.  Personally, and I think I remember reading somewhere, the general consensus is that around 10-20 MJ is the boundary between a gas giant planet and brown dwarf, but I think it's fairly arbitrary, much like what's a planet vs. minor planet, Kuiper belt object (KBO) or asteroid.

So during Solar System formation, was there a chance Jupiter could have been a star and it failed ("failed star!") because the mean Sun gobbled up all the mass? Not really, at least not in our solar system.  But for getting the very basic concept across of going from a gas giant planet to a star, calling Jupiter a "failed star" can be a useful analogy.
