Yes you can legally be denied a public defender even if you were arrested and bonded out of jail.
Most times you'll be appointed a public defender or court appointed attorney if you cannot afford one on your own.
Although a public defender or court appointed attorney is not as good as a privately paid for attorney or lawyer who is really interested in your case.
If you're facing jail or prison time then most times you'll be approved and provided a public defender but if it's a case where you're not facing jail time or prison time then you can be denied a public defender.
It's best though to hire a private attorney on your own who will fight for your case more because they will be interested in your case more than a public defender would be.
Also for municipal court cases a public defender is usually not provided although I believe that everyone should have available to them a public defender no matter what the case is because we all pay taxes.