It only takes a counter-example (an example of a shorter program) to show that an answer is incorrect, but you could probably pick up a Turing Award or two if you find a way to put a tight bound on the Kolmogorov complexity of a bit string.
It's only difficult in general. Nobody says there aren't any easy instances.
Sure, there's a bunch of trivial/easy ones for programs that do almost nothing, but I really wonder if are there any interesting programs for which tight bounds are known?
Even if we limit it to primitive recursive functions, how can we put a bound on the Kolmogorov complexity of one? We'd need some way of coming up with bounds on the smallest Turing machine to compute that function.
That might be possible. Primitive recursion is much less powerful than Turing machines.