Your assembler might be talking about component height for the Pick-and-Place handler. Some modern assembly machines use a vision system that measures reflections, shadows, and contrast to allow the machine to "see" the PCB and its components. As one example, Juki makes such a machine ( **broken link removed** ).
He might also be talking about "IR shadowing" where taller components cause non-uniform heating by blocking infrared heating sources used for reflow soldering. That problem is generally solved by using forced convection heating instead of direct IR.
The fab is only concerned about stubs because the CAM software does a check for them. They are acting as a backup for you when they check the Gerber files before they produce the photomasks used for PCB etching. The presence of a stub isn't a problem for fabrication - they just make you aware of it so you can fix it if you want. Over the years, I've had PCB fabricators catch shorts, opens, stubs, drill size problems, etc. They can be your best friend when it comes to checking for mistakes you may have missed. That's why it's a good idea to establish a good working relationship with a fab you can trust - they work as part of your team.