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.
Best Answer
The smallest objects (given an elemental abundance mixture appropriate to a giant planet like Jupiter) that can attain hot enough interiors to ignite a sustained thermonuclear reaction are about 13 times that of Jupiter. The fusion reaction in question is that of deuterium, which burns at lower temperatures than "normal" hydrogen. The equivalent mass for hydrogen (protium) ignition is about 75 times the mass of Jupiter.
Jupiter is therefore nowhere near massive enough to instigate nuclear fusion reactions in its interior.
The question arises - why cannot Jupiter simply contract until its core becomes hot enough to ignite these reactions? The virial theorem tells us that any contraction will be associated with half the gravitational potential energy being radiated away and half being used to heat the interior.
The answer is degeneracy pressure. The electrons in Jupiter's deep interior are dense enough to form an increasingly degenerate Fermi gas. This Fermi gas exerts a pressure that is almost independent of the temperature. This means that as Jupiter radiates away its interior heat, the pressure hardly diminishes and as a result any contraction of Jupiter is rather small and slow and will not lead to any significant temperature increase. Ultimately, Jupiter will attain a "zero temperature" configuration that is not much smaller than it is today and could cool at nearly constant radius. At no point will it ever become hot enough to ignite nuclear fusion. This is the basic reason for the mass limits quoted above.
I found the following plot in Fortney et al. (2006) which shows this general behaviour. These are model calculations of the evolution of radius (in Jupiter radii) for a Jupiter-mass planet either with (dashed lines) or without (solid lines) a solid core. The different colours represent different distances from the parent star (to account for the effects of insolation). The appropriate curve for Jupiter is between the 1 au and 9.5 au curves. What this shows is that the rate at which the radius gets smaller decreases with time (note the time axis is logarithmic).