Sure it could be the corks. I see 3 possibilities.
1. The corker.
2. The corks. Have you used this brand before? Is this a new or old batch of corks? [BTW, I do not recall seeing corks pieces in the bottle, and I ran a Ferment on Premises for 6 years, and used a few brands of corks before settling on a preferred one.]
3. Your process. Specifically, soaking the corks is the wrong thing to do. This removes a coating on the corks. [But, I don't think it causes the corks to break up. However, since I don't do this, I don't really know.]
Just remembered something. Is it many small pieces per bottle or one small chunk out of an edge of the cork? If one small chunk, I'm pretty sure it's the corker.
Suggestion, pull a cork out of a bottle and post a picture here.
Steve