Theoretically, we could get very very close to the speed of light, but not quite reach it. If we were in a spaceship and had an infinite amount of fuel, we could carry on accelerating and getting faster for ever.
But not quite for ever, because as we get closer and closer to the speed of light, we get heavier and heavier, so there is more mass to accelerate, and so we accelerate more slowly, and will never quite reach the speed of light (this is what Einstein discovered, it is part of his theory of relativity).
Of course, to even get close to the speed of light we would need to accelerate for a long time, which would require an enormous amount of fuel. There have been suggestions of how we might cope with this problem though. For instance, the spaceship could be attached to a enormous ‘solar sail’, like a ship sail but hundreds of kilometres across. This would catch the photons, or light particles, emitted from stars and use their energy (very little for each one, that’s why we need a big sail), like a ship’s sail uses the wind, to propel our spaceship along
Awesome answer by John… he has way more physics knowledge than me!
I would say that in the future anything is possible. We are always making major advances in science and always breaking barriers we never thought we could break! Right now its impossible to reach anywhere close to the speed of light but 100 years ago people would have thought it impossible to go to the moon and 50 years later they did!! So I reckon in time we could get really close or even match it – you never know!