It is possible. It is the judge's choice whether you will get a public defender. Essentially, the defendant requests a public defender and then the court must determine whether the defendant can afford to hire a private attorney or whether the defendant is indigent.
Juan C. Garcia, Jr.
GARCIA LAW OFFICE, LLC
Being appointed the public defender is dependent on whether or not you are indigent. The court has requirements that they follow in determining whether or not an individual meets the financial criteria for being indigent. Many courts will require you to complete a financial declaration that is signed under penalty of perjury. In determining whether or not you qualify for the Public Defender the court is entitled to enquire as to how you were able to post bond.
Often this is different from county to county. My experience is that the smaller the county, the more likely it is that the judge will require you to hire your own attorney if you have posted significant bail.
Depending on whether you are deemed indigent by the judge you are before, is whether you qualify for a public defender. Posting a bond tells some judges that you are not indigent but may allow a public defender to be appointed after a hearing. The judge may allow you to use the bond to be applied toward attorneys' fees. Some judges even have a bond referral list of attorneys who will work for the bond and not charge more than the bond posted.