Old Cheap Dude's suggestion is a solid winner for the right situation. But, there a couple of options, no, the entire fence wouldn't need to be replaced. You either need to re-install the existing posts deeper (without the concrete silliness) or to add posts between posts. Concrete's use usually means installing the fence posts shallow, it's always been a dumb practice.
- The soil's just simply too wet and loose for the posts to handle the boat-sails between them. As an addition to the above, you can add a larger than the post Pan/Foot or even "T" the bottom of the posts with a piece of Wood or Heavy Aluminum (diamond plate works well) to dramatically improve wind-shear strength.