Never
Someone asked me to estimate how long it would be until we developed a hyperdrive.
I am bold enough to say: Never. We will never break the speed of light. Never.
Why so confident? Because when Einstein proposed his theory, he did not abolish Newton. Apples did not start falling up.
At slow speeds and low temperatures, Newtonian mechanics explains everything from the fall of apples to the flight of cannonballs to the motions of the moon.
When Copernicus proposed a revolutionary model of the universe, sun-centered, based on elliptical orbits, the sun did not start rising in the west; nor did the discovery that Venus had phases like the moon make the Morningstar able to rise to the zenith.
The observed behaviors of the universe do not change when science evolves to a new level: all that happens (and it is a wondrous happening) is that anomalies can now be explained, that one explanation can now be used where two or more ad hoc explanations previously had been.
So here. The Einsteinian model of the universe is too robust. It explains too many otherwise scattered observations, from the behavior of particles in accelerators, to clocks going out of synch on satellites, to the procession of the axis of Mercury. The information is too solid.
The next evolution in science might harmonize quantum mechanics and classical relativity. A revolution in philosophy might solve the mind-body problem. But unless our theory of how the material world works is wrong, wrong down to its roots, wrong to the point where everything studied since the time of ancient Greece is wrong, you are not going to take matter that has mass, and accelerate it to light-speed, using less than infinite energy.
We can hope that science finds some quirky exception to the rules somewhere, or discovers that matter and energy, time and space do not behave in some circumstances as they do in every circumstances observed so far, but, so far, that hope has nothing serious on which to feed.
It is not like developing heavier than air craft when you see birds every day. It is like trying to design a thermometer that will measure temperatures below absolute zero: something which, unless our basic theory of how things work is wrong at a basic level, our theory calls impossible.