Is a DDR3 MB compatible with DDR2 RAM?

If a motherboard's specifications say it supports DDR3 SDRAM sticks, would I be correct to assume it is also compatible with DDR2 RAM? Am I wrong, or does it depend on the motherboard?

No. DDR3 and DDR2 are designed differently. Both have different key placing so they won't even physically fit, not to mention differences in electrical voltage:

There are motherboards which will provide entirely separate slots for DDR2, but you cannot use DDR3 in DDR2 slots, or both types together.

Supports DDR3 SDRAM sticks, would I be correct to assume it is also compatible with DDR2 RAM?

That's incorrect. DDR3 Rams are PIN & voltage incompatible.

Or does it depend on the motherboard?

Certain motherboards provide DDR2 & DDR3 slots, but you can use only 1 type at a time, you cannot mix and match both