4 Layer PCB stack up
The problem is indeed with the crossing 1-15 and 2-16 vias. The manufacturer has to be physically able to make the board. There may be a way to attempt that construction, depending on the equipment the manufacturer has.

There are two straightforward ways to get close to what you have drawn.

1) 1-2 core, drill, thru plate. 15-16 core, drill, thru plate. Assemble, drill, thru plate. That gives 1-2, 15-16, 1-16, but not 1-15 or 2-16

2) 1-2 core, drill, plate. Prepreg 15 foil, drill, plate, prepreg 16 foil, drill, plate. That gives you 1-2, 1-15, 1-16, but not 2-16 or 15-16

More expensive, but I have used the process, micro-vias. It's mainly used for their small footprint, to get tracks between BGA balls.

3) As for (2) but with a final micro-via step which is laser drilling 15-16, power set to ablate the prepreg, but stop at the 15 foil. These vias would be very small and fragile, pads connected to them do not generally survive rework.

4) Speculative. The micro-via process only works because of the small thickness 15-16. It may be possible to micro-via from 16 through to 2, via an aperture etched in 15. Ask the manufacturer. They may have the capability. If you wave money at them, they may be able to develop the capability for you. 

5) Speculative. You could replace the 2-16 via with 2-15 and 15-16 micro-vias, if the board vendor was happy to insert another micro-via step in the (3) buildup above.

It would be far better to think harder about your layout, and try to use the standard routes.

Do note that your board construction is unbalanced through the Z direction, this is strongly discouraged by most board vendors. It will probably warp after exposure to soldering heat, cracking your ceramic caps.