Short answer. Yes. Any adverse party may contest the grant of that authority at any time before the trial court renders judgment, so there is no early window that closes. If the challenge succeeds, the clerk of court assesses and collects the proper docket and other lawful fees.
What the law says
Any adverse party may contest the grant of such authority at any time before judgment is rendered by the trial court.
Rule 3, Section 21 — Indigent party. Read the full provision →
The challenge stays open almost to the end
Most objections in civil procedure must be raised at the first opportunity or they are waived. Indigency is deliberately different. The rule lets any adverse party contest the grant at any time before judgment is rendered by the trial court, so a defendant who said nothing when the authority was first given can still raise it months later, after discovery has shown what you actually own. The practical effect is that your declared circumstances remain open to scrutiny for the whole life of the case at that level, and you should expect them to be tested if the other side has reason to think you have means.
What the court is measuring
The standard in the rule is concrete rather than impressionistic. The question is whether you are a party who has no money or property sufficient and available for food, shelter and basic necessities for yourself and your family. Two words carry most of the weight. Sufficient means the comparison is against basic needs, not against the cost of the lawsuit, so being unable to afford a long case is not by itself the test. Available matters where an asset exists but cannot be turned into cash, since property that is already encumbered is not the same as money you can spend on fees.
If the challenge succeeds
The consequence is not dismissal of your case. If the court determines after hearing that the party declared as an indigent is in fact a person with sufficient income or property, the proper docket and other lawful fees are assessed and collected by the clerk of court. You then have whatever period the court fixes in which to pay. If payment is not made within that time, execution shall issue for it, and the rule preserves such other sanctions as the court may impose. So the realistic downside is a sudden bill, plus a credibility problem in front of the judge who will decide your case.
Preparing for it
Because the exposure runs for the whole case, keep the documentation current rather than filing it once and forgetting it. If your circumstances improve materially while the case is pending, through a new job, an inheritance or a sale, that change is exactly what an adverse party will surface. Raising it yourself is usually better than being caught with it. The court already has a record of what you claimed, since the rule requires an ex parte application and hearing before the authority is granted at all, and it is inconsistency with that record that does the real damage.