Short answer. The appeal can be dismissed. That the order or judgment appealed from is not appealable is expressly listed as a ground for dismissal, and the court may raise it on its own motion. Nothing you do afterwards in the appeal cures that underlying defect.
What the law says
(i) The fact that the order or judgment appealed from is not appealable.
Rule 50, Section 1 — Grounds for dismissal of appeal. Read the full provision →
A ground unlike the others on the list
Most of the grounds in this provision describe something the appellant failed to do — a late notice of appeal, unpaid fees, a brief not filed, steps not taken to complete the record. The last one is different in kind. It concerns the thing being appealed rather than the appellant's conduct: the fact that the order or judgment appealed from is not appealable. No amount of diligence cures it, because the problem is that there was nothing to appeal in the first place.
Why some orders cannot be appealed
The general scheme is that appeals lie from judgments and final orders that dispose of the case or a distinct part of it, leaving nothing further for the court to do. Orders that merely resolve something along the way leave the case alive in the court below, and allowing each of them to be appealed separately would fragment a single case into many. The remedy for that class of order lies elsewhere in the rules, not in an ordinary appeal.
What dismissal on this ground actually costs
The dismissal is discretionary in form — an appeal may be dismissed — but there is little to exercise discretion on where the order genuinely is not appealable. The real cost is time. While the misdirected appeal runs, the period for the correct remedy is usually running too, and it is often much shorter. That is why this defect is worth identifying at the outset rather than after briefs have been filed.
Checking before you commit
Read the order and ask what remains for the court below to do. If the case continues there, be cautious about treating the order as final. Look also at how the order is framed: a ruling that disposes of one claim entirely may be appealable as to that claim even though other claims continue. Where it is genuinely doubtful, the safer course is to identify the correct remedy and its period first, because the choice is difficult to unwind later.
Who can raise it, and when
Because the defect concerns the order rather than your conduct, it is not cured by the appellee's silence, and it can surface late — the court may act on its own motion, and nothing in the rule requires the point to be taken at the outset. That is the practical risk: an appeal can proceed for months on a foundation that was never there. If you suspect the order is interlocutory, raising the question yourself and seeking the correct remedy promptly is far better than discovering the answer in a dismissal.