Can you tell from the periodic table exactly how many neutrons are in an atom?

Yes

Explanation:

On the periodic table, you can find the element symbol, name, atomic number, and atomic mass. Just knowing the atomic mass and atomic number can give you the amount of neutrons. The atomic number states the amount of protons. The atomic mass is proton+neutron. Knowing this, you can subtract the atomic mass by the atomic number to get neutron.

Example: Carbon

Atomic number: 6

Atomic mass: 12.011 or 12

12-6=6                                                   [Atomic mass - Atomic number]

There are 6 neutrons in Carbon.