With any normal liquid fuelled rocket engine it is possible to throttle the thrust up and down, but in Kerbal space program it is possible to bring it all the way down to around a percent thrust. Is this possible with normal rocket engines?
It is not usually possible to throttle a rocket engine that deeply for a range of reasons and especially for liquid propellent rockets. As an example the merlin engine of the Falcon 9 booster provides so much thrust and the booster has so little mass as it approaches the landing pad, that it is very hard to get it landed in one piece at all because even at minimum throttle there is too much thrust (it can't hover).
The solution adopted by SpaceX, rather than attempt to adjust the engine for deeper throttling, was exact timing. As the booster descends with the engine firing it is losing speed. The landing is timed so that at 0 altitude it is at zero velocity and the engine cuts out.
If the engine cut out any sooner the booster would crash into the landing pad under the force of gravity any later and the booter would take off again.
Typical rocket engines can't throttle that low, but it's possible.
The widest throttle range I know of is the sustainer engine of the Lance tactical missile, at 350:1 (yes, really), meaning it can be throttled to about 0.3% of full power.
The Space Shuttle main engine could throttle down to 67% or so, and the Falcon 9's first-stage Merlins are claimed by some to throttle down to 40%.
Widely-throttleable engines tend to run into problems with stable combustion and exhaust flow, so it tends to be easier to design engines with little or no throttling capability.