@NathanielStenzel Is it PLA, I believe. What are the alternatives you are talking about? What is your heated bed typical temperature when it fails to print? Would it typically happen when the model is [too] tall?
I like a 100C heated bed, but I print ABS. For ABS, an alternative is ABS mixed with acetone (the mix is called ABS juice) dried on the bed. I think PLA users sometimes use blue painter’s tape as a substitute for a heated bed.
The last time I had delta failures (layer shifts) at a particular height, I found the bowden tube was binding up on one of the carriages. I kept missing it because it happened far enough up the print that I’d walked away to do something else.
This probably isn’t the problem you’re having, though.
You need some meatballs with that spaghetti!
Could you post a picture from side on and give an idea of the object size? it’s possible you are experiencing several issues that give the results you are experiencing.
@David_Selwood David, first pic is letter M, here is the bottom view: https://photos.google.com/share/AF1QipN1PbyuNKweu8MsHXo_yg0UVYfSNJJUI7KugDFKWlZ5PWvu8vsk1wY6WUv58-s5qQ?key=Z2pjN2o0SE9PX0tFMEhHRzMtOVI0MFJ6Sl9NTGpn
Here is a link to the whole album: https://photos.google.com/share/AF1QipN1PbyuNKweu8MsHXo_yg0UVYfSNJJUI7KugDFKWlZ5PWvu8vsk1wY6WUv58-s5qQ?key=Z2pjN2o0SE9PX0tFMEhHRzMtOVI0MFJ6Sl9NTGpn
The second pic is just the test cube 25x25mm(very roughly it is about 1’x1’') obviously unfinished(shorter/unfinished side is ~8mm).
Layer height is 0.5mm
cube printing attempt was to confirm dimensions, angles, etc. The letters were printed just for fun/testing with different speed/temp/fill factor, etc
Thanks, I would suggest that you make sure everything is tight and that there is no play in your print bed, extruder or X-Y movement (ie no wobble in the carriage before you can slide it). Is your slicer software configured for the correct dimensions? And lower layer height down by half with a reasonable heat and speed. I print PLA at 193° @ 40m/s with no heated bed (on blue painters tape and a thin spread of copydex glue, dry before printing) If none of the above resolve or help you then i’m afraid i’m outside of my area and maybe some of the other suggestions here can assist.
@David_Selwood Thanks a lot!
If you suspect the slicer/gcode, just give it a look with gcodeviewer for example. This would be extremely surprising though (note that you can also grep ‘G[01].*Z’, extract the Z values and plot them to check how they are spaced).
If your bed is moving down, double check the couplers and/or pulleys! Well, even if it does not 
If people would please look at the bottom of image #2, you will see curling where it lifted off of the bed.
I got it now - the motor drivers were overheating. Documentation was wrong, 1.2v is too much for the electronics. Lowering voltage to 1v helped a lot - cube is printed: https://photos.google.com/share/AF1QipOwn5aCA1PMgcoimmxosinNBpJRm1d6syQ6BdxFMIWjSK-o7Et1_wJNNf8hJWB3sA?key=LUI2V1huQUNYbkMwdndGb3ktZm10NGJvRUZfVWpR 
Although, I might set the votage even lower, as the heat sinks still feel pretty hot after the print. Special thanks to @NathanielStenzel , @Tim_Visible and @ThantiK and everyone else
@Vera_Dobrianski interesting, so overheating meant you had larger Z steps? Sounds weird to me! Nice guess, @ThantiK !
@Jeremie_Francois I have a delta printer, so all 3 motors are involved when changing Z (or X or Y, in general case). So, I think what was happening - one motor was overheating more (or earlier) than other two. Missing one step on A(or B or C) would completely mess up the nozzle position.
Note - it was not doing the jerky movements as one of the videos in this thread shows. The nozzle would typically start going up (and off on X-Y), and after a while would come down and may be even lower then the layer it was printing. I do not have a video of it, but the behavior is very recognizable for me now.
@Vera_Dobrianski oh it totally makes sense then (my delta is just so idle on its shelf that I forgot about this -hellish- exquisite geometry here), thanks!
@Vera_Dobrianski I see you still have lifting. I am glad to hear you have the motor overheating fixed. One tip for the variable resistor ( aka pot aka potentiometer ) driver is to listen to the motors. I also recommend listening to it print and for that case listen to food as you cook it, but that is another matter.