z-axis hard to move in some areas - what could be faults, how to improve? I have a z-axis which follows schematically the same principle as the makerbot one's (threaded rod and two leading rods with linear ball bearings carrying a level). It is from an old experimental lab doing physics or chemistry.

The axis move gorgeously about 5 cm, but then it get's stuck on either sides of this way. Both driving threaded rod and the leading rods seem perfectly fine and should be able to allow for further movement. This is as far as I can see by eye.

Where should I look to find further issues and how could I improve the performance? Do I need to take the construction apart?
X stage binding like this is almost always caused by parallelism issues with the rods and/or screw. The two-rod-plus-screw arrangement is quite over-constrained and thus requires good alignment to move smoothly. 

Some basic troubleshooting steps:


Make sure the screw is not constrained at both ends. It is very difficult to manufacture a screw that is perfectly straight and then mount it so precisely that it can rotate with zero runout. Screws in light-duty linear motion applications should be allowed to "wobble" freely so any bend or runout doesn't apply side-load forces to the Z stage. This can be accomplished by putting a misalignment-tolerant coupler (like an Oldham coupler) on the driven end, and/or leaving the non-driven end free without any support bearing. A motor mount with a small amount of compliance (like a rubber stepper damper) can help when the screw is captive to the motor. 
Lubricate the screw and bearings and make sure everything is clean and in good condition. 
Allow the Z carriage to self-align the rods and screw. The proper technique for this will depend on the Z stage, but the basic idea is to loosen the rod and screw mounts on either end, run the Z carriage back and forth a few times to push the rods into position, and then only re-tighten the top and bottom hardware when the carriage is at that end. It may also be necessary to loosen the screw nut and bearings on the carriage to get everything aligned properly and smooth-running, but that does not enforce parallelism like loosening the rod and screw mounts, so is really a secondary step. It may be necessary to leave some "float" in one rod or the bearings on one rod (with gravity preloading out any resultant slop) if the hardware has major alignment issues. 
If the stage still binds after doing the above, it may be necessary to check if the linear hardware is bent (such as by rolling rods on a flat table) or use a file or Dremel to loosen up the mounts or whatever feature is causing the misalignment.