I have been reading about Jupiter as of late and reason I am asking is if Jupiter has strong enough gravity to pull hydrogen to itself then it should be only matter of time when it will acquire enough mass to ignite itself and become a star, same would apply to any other planet that is big enough to hold hydrogen (In other words if planet gains more matter than it looses it will become a star eventually). Or am I missing something here?
Jupiter will never (not on any timescale like the lifetime of the Sun anyway) accrete enough mass to begin hydrogen fusion.
It would need to accrete 12 times its current mass to undergo a brief period of fusing its interior deuterium and to accrete more than 70 times its current mass to attain a central temperature high enough to sustain hydrogen (pp chain) fusion. There isn't enough accretable mass in the solar system to do this.
The problem is electron degeneracy pressure, which is able to support lower masses even if the gas is cooled completely. The planet cannot contract and heat up its core enough to begin fusion.
I'm not entirely sure what you mean about 'pulling hydrogen', all bodies, whether they be planets or literally human bodies, will pull hydrogen via gravity. Earth can lose the H it attracts as H is so light that it can have speeds greater than the escape velocity (just due to random thermal motion). Perhaps Jupiter is sufficiently massive that this happens significantly less.
However having hydrogen isn't the only condition needed for a star to be born. That hydrogen needs to be hot enough that it is moving fast enough that it can collide with another hydrogen atom at sufficient speeds to overcome electromagnetic repulsion and get close enough for the two to fuse, releasing energy. These temperatures are only reached in incredibly dense high pressure cores of massive collapsing clouds (and the stars these subsequently form). There is a sharp cut-off between the temperature, and hence mass needed, for fusion to start, and Jupiter falls slightly short. (The cutoff is at about 0.08 solar masses)
If jupiter were a bit heavier deuterium fusion might occur, giving a brown dwarf "star", but, happily for us, it'll just while away it's life as a gas giant. Binary star systems, where two bodies in the system are massive enough to form stars, are fairly commonly seen in the galaxy (and more than two bodies is perfectly possible, but the system is unstable and all but two will be ejected pretty quickly) but it's probably a good thing for life on earth that we're not in one!
Can Jupiter become a star? Yes and no.
Jupiter can never self ignite because it lacks the necessary mass. Hundreds of Jupiters would have to be combined to produce enough gravity for self ignition.
However, Jupiter can be ignited by a high energy device such as a nuclear bomb. In a way that would make it a star for a very short time. A star like the sun has a careful balance of gravity vs expanding forces due to temperature. Jupiter lacks the gravity to oppose the expansion forces and hold it together. In all likelihood igniting Jupiter would be more like blowing it up. It would be the biggest hydrogen bomb ever observed. Parts of Jupiter and it's moons would rocket through the solar system combined with a giant flash or energy. The effects of such an explosion could be devastating to Earth. I would speculate that half the Earth facing Jupiter would be instantly scorched by the incoming energy. The temperature of the atmosphere would go up all over the globe. We would have giant atmospheric events, leveling ground structures and forests. Even the oceans and land would get warmer. And that's when the meteor storm would start. Pieces of Jupiter would start hitting Earth. Best case, we'd get away with half a dozen craters from the bigger chunks while most burn up. Each impact would be devastating, eclipsing nuclear weapons. Worst case, a big chunk hits us, ending life on Earth.
This could make for a great Hollywood disaster movie.