It does not stack.
Start of turn 1
Jirachi used Stealth Rock!
Pointed stones float in the air around [VP]The Player's team!
The foe's Smeargle used Bind!
It's not very effective...
Jirachi lost 3 HP! (0% of its health)
Jirachi restored a little HP using its Leftovers!
Jirachi was hurt by Bind!
Start of turn 2
Jirachi used Stealth Rock!
But it failed!
The foe's Smeargle used Wrap!
It's not very effective...
Jirachi lost 3 HP! (0% of its health)
Jirachi restored a little HP using its Leftovers!
Jirachi was hurt by Bind!
[Mew]: kk, thanks.
[VP]The Player: Nope doesnt stack
Tested on PO.
Confirmed: It does NOT stack. I repeat, it DOES NOT stack.
I tested in on my White today with a Shellder that had Clamp and Whirlpool. I used Clamp first, and then Whirlpool. Only Clamp's effect continued.