If USCIS does not have enough information to approve your application for naturalization, they will send you a 30-day notice to provide evidence of eligibility or be denied. Here's an overview of what you can do next.
Read the notice of intent to deny several times. Outline the issues and responses. Identify any statements that are factually incorrect.
USCIS officers meet with a lot of applicants and they may get their facts wrong. Be sure to address any errors on the notice and back up your statements with evidence. For example:
Tie the evidence together with sworn statements by you or another person with personal knowledge of the events in question with an affidavit. Do not use sworn affidavits in place of evidence.
Make sure USCIS receives your response on or before the deadline, even if you don't have all of the evidence you need.
Usually, USCIS will request evidence that you simply cannot get within 30 days. You might be waiting on birth certificates, marriage certificates, death certificates or other civil registry documents from a third world country. Don't delay. Ask your family member to scan or fax it to you before mailing.
If evidence is coming that you don't have by the deadline, submit it afterwards. It is not ideal, but USCIS will see that you responded on time. Hopefully, they will not make a decision on your case until you have had a chance to submit all of the evidence.
Deliver your response using a method of delivery that can be confirmed. Our firm only uses FedEx to deliver client applications, responses to requests for evidence, responses to notices of intent to deny and appeals. US postal service delivery can be inconsistent in getting you back that silly green post card. If your NOID indicates delivery to a PO Box, find out the street address and send it to the street address. We have found that PO Boxes are not picked up daily by USCIS and can cause a delay in the processing of your application.
If you cannot send sufficient evidence to USCIS, your application will be denied. From there, you have a few options
Appeal the decision. USCIS will send you a denial letter along with form N-336. Complete this form to request an appeal if you believe USCIS made a mistake.
Re-apply when you become eligible. If there was an issue affecting your eligibility, you can usually wait until enough time has passed that your record has cleared. The 5 years leading up to the time your application is filed are the most important to USCIS. For example, if a missed a child support payment from 2013 lead to your application being denied, wait until 2017 to re-apply.
Keep in mind that if you've had issues with eligibility in the past, it may be harder to re-file or appeal your case. An experienced immigration lawyer can review your case history and provide solid advice about what to do next.