When may an alien not be eligible to receive an immigrant visa despite an approved petition?

The eligibility for an immigrant visa, despite having an approved petition, is closely tied to the priority date of the application. When the priority date is current, it means that an immigrant visa is available for the applicant based on their petition category and country of origin. Therefore, if the priority date is not current, it indicates that the applicant must wait until their priority date becomes current before they can proceed with obtaining an immigrant visa.

In the context of immigrant visas, the availability of visas is governed by the Visa Bulletin, which outlines which priority dates are currently being processed. If the priority date does not fall within the dates listed as current, the individual is ineligible to receive a visa, even if their petition has been approved. This situation emphasizes the importance of matching the date to the current processing capabilities.

The other options refer to situations where eligibility would typically still be intact, as long as the priority date is current and a visa number is available. Thus, the reasoning behind the correct answer is grounded in the critical role of the priority date in the immigrant visa process.