This one?
Inputs are spec'ed at 1uA, so 16 connected in parallel is only 16uA, and 128pF of capacitance. Easily driven by '328 outputs.
I would suspect some miswiring somewhere.
How are the parts powered? 16 parts would need 336mA under no-load conditions. If you're powered from USB, you may be hitting the 500mA limit USB will supply when you add in the rest of your system.