This would not be true under the conditions and technologies we currently possess. It's as simple as asking any astronaut; if time passed in such a way, astronauts would appear to be gone for dozens of years before returning, but this is not the case. If we ever attain the technology that allows us to travel closer to the speed of light, we would begin to see effects like the one described in the question.
The scientific process behind the story you've heard is a phenomenon called time dilation, which is a side effect of relativity. The passage of time has been shown to be influenced by a variety of factors—particularly speed. Scientists have put clocks into orbit, compared them with clocks on the ground, and found very small—but measurable—differences between them. These differences become exponentially larger as you near the speed of light. This is described by the Lorentz factor, which can be calculated mathematically as
where v is the velocity of the moving object and c is the speed of light.
Then, we can compare the time dilation, or delay, between two relative events by applying the Lorentz factor to it. This can all be simplified to the following equation:
t = t0/(1-v2/c2)1/2
Where t = the elapsed time, t0 is the observed time by the traveler, v is the traveler speed, and c is the speed of light.
We can rearrange to solve for v:
(t0/t)2 + (v2/c2) = 1
Using the values provided above (t0 = 3, t = 400),
v = 0.99997c