The showstopper for Earth launch is the atmosphere. Drag grows proportional to velocity squared, so materials accelerated up to several km/s are going to slow down very quickly. Each doubling of velocity is only going to give you the same amount of extra jet range as the previous doubling. This is a very efficient ceiling for how fast things can go inside the atmosphere.
But looking more abstractly at it beyond Earth launch, there's nothing wrong with this idea, the only limitation being relative velocity.
If the craft is travelling faster than the jet hitting it, there's no acceleration. And even before that, this has diminishing returns due to the relative velocity between the jet stream and the craft approaching zero.
So how fast can we make those jet streams?
- Conventional chemical reactions limits you to 4-5 km/s. Note how this isn't even enough to reach orbital velocity.
- Devices based on the heating of gas have velocities limited by how high a temperature they can stand. Around 15 km/s at most, and reactors probably limited to about 10 km/s.
- Electric and magnetic ways of accelerating a jet could reach a 100 km/s and more, but can only accelerate a very small mass flow.
- Even light carries momentum, so shining a torch at the craft theoretically has no acceleration limit, even if the thrust is absolutely minuscule. (Efficiency is still lost to red shift as the relative velocity gets very large).
There's no "maximum distance" as such, there's no problem waiting years for a jet stream to catch up to you in space.
The idea of accelerating a bomb up to you has a more fundamental flaw: If we have a way to make the bomb reach such a velocity, why aren't we using that for our craft instead? If the answer is to use bombs to accelerate the bombs, that's what the rocket equation is about in the first place. You save no mass by doing this, but introduce the complication of fragmenting your "rocket" into many different parts.