NO, it should not happen. That is the whole purpose of a drop-away rest---fletching clearance.
Who set it up? It's possible, and most likely that the rest is not timed correctly. It should not raise up to it's highest level until about the last
1 1/2" of the draw cycle. And yes, it's fixable. I'm not sure about any adjustments on your particular rest, but maybe you could slide the cord attached to your down-cable up a little bit. If it's got serving around it you still might be able to work it out. Sometimes it only takes 1/8" or so.
The other two possiblities are the nocking point could be too low or your arrows are underspined, bending too much as they leave the bow. OOPS, does the rest have an adjustment for the spring tension? If so maybe you could put more tension on it to make it drop quicker.
You need to ensure that you have your center shot setup properly, that is; when you set up, adjust your rest so that the center line of the arrow is centered in the berger button and adjust your nock so that the arrow sits perfectly level, as a starting position, then fine tune by paper tuning. it's important that you have straight nock travel, if not the rear of the arrow can get kinda snapped down on the rest as its leaving the bow, resulting in contact. if your bow is perfectly tuned, then the rest is simply not getting out of way fast enough. like fisher said, make sure the rest is reaching its full upright postion, the last 1" -1.5" of your draw stroke.
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