Can I stop the Department of Education from garnishing my paychecks for student loans?

Yes.

It might be possible to stop it administratively, by requesting a hearing after you receive your required notice that you are about to be garnished. You can challenge the garnishment for certain reasons, including that the garnishment will cause financial hardship to you or your dependents. 

If you don't win your hearing, and you qualify, filing chapter 13 bankruptcy stops the Department of Education from garnishing your paychecks. Federal student loan garnishment can otherwise result in 15% of your disposable pay being deducted and sent to the government.