Say a person is granted indulgence after Saturday night mass, (which fulfilled the Sunday obligation). Does that mean he can not be granted indulgence the following morning?
I'm assuming an indulgence can only be granted once a day.
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Can an indulgence be granted more than once a day?
It all depends if it is a plenary indulgence or a partial indulgence. Only a plenary indulgence may be obtained once a day. Partial indulgences may be obtained as often as one desires, as long as all the conditions are met.
An indulgence that can remove all the temporal punishment due to forgiven sin. No one but God knows for certain when a plenary indulgences is actually gained, because only he knows whether a person's dispositions are adequate. One norm for such dispositions is that "all attachment to sin, even venial sin, be absent." If these dispositions are in any way less than complete, the indulgence will only be partial. The same provision applies to the three external conditions necessary to gain a plenary indulgence: sacramental confession, Eucharistic Communion, and prayer for the intentions of the Pope. If these conditions are not satisfied, an otherwise plenary indulgence becomes only partial. These conditions may be satisfied several days before or after the performance of the prescribed work, though preferably Communion should be received and the prayers offered for the Pope on the same day as the indulgenced work. A plenary indulgence can be gained only once a day. - Catholic Culture
As to your question about receiving an indulgence on Saturday evening and then another plenary indulgence the next morning. "Does that mean he can not be granted indulgence the following morning?" A plenary indulgence may be obtained only once a day, that is to say from midnight to midnight of the same day (January 1st).
We have some parishes within our archdiocese that make it possible to obtain a plenary indulgence by reciting the Te Deum just before midnight on December 31 and then another one just after midnight on January 1 for the recitation of the Veni Creator. The anticipated Mass on Saturday has no bearing on receiving an additional plenary indulgence the following morning.
A partial indulgence is granted to the faithful, who devoutly recite the hymn Come, Holy Spirit, Creator blest.
A plenary indulgenceis granted, if the hymn is recited publicly on the 1st of January and on the feast of Pentecost.
A partial indulgence is granted to the faithful, who recite the Te Deum in thanksgiving.
A plenary indulgence is granted, if the hymn is recited publicly on the last day of the year.
The official Vatican list of plenary indulgences can be read here, however it is in Latin.
The only day one is able to receive an additional plenary indulgence on the same day is on one's deathbed (In articulo mortis).