yes. you will always be able to find it, its just about if you're lucky enough or not.
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beat the league again and it will be roaming again
Suicune, if i do remember correctly, runs off to the area right of Bills house north of cerrulian city (pardon my spelling)
No You dont
you dont need badges you chase suicune then you battle it were you meet misty if you fail yo get another chance to catch in burned tower
It depends on the Pokemon game you're playing. In the 4th generation and later games, you can re-encounter Suicune if you defeat the Elite Four again, while in earlier games, you won't be able to encounter it again.