Thanks for your attention!
1) Pen made for the Surface Pro 4Okay.
Next step is to isolate a pen hardware problem.
1) Are you using the pen made for the Surface Pro 4/Surface Book? This is the Surface Pen (V4), not the pen with two buttons near the tip.
2) Do you have a friend or acquaintance with the same pen? If so, it would be very beneficial to ask that friend if you could temporarily pair their pen with your Surface Pro 4. You would have to in-pair your pen first, then pair their pen and test. Of course, this is inconvenient, but very helpful to avoid unnecessary worry about the computer or pen that you own.
So I decided to put my pen to the test since I just got a refurbished Surface Pro 4. It has Windows Version 1607 (OS Build 14393.969) all updated etc. I opened Microsoft Word and went to town with the pen. Initially the pen did skip intermittently, so I kept on drawing / spelling out letters etc. I then closed Word without saving and reopened it and did the same thing but this time the "skipping" went away and I got fluid lines without any breaks.It creates breaks in the lines drawn in every program. (OneNote, FreshPaint, Sketchbook, etc).