Can you hike Half Dome on the day the cables come down?

Yes. The National Park Service defines the cable season to include the day they come down, not to end before it. NPS states that the cables "usually go back up the Friday before the last Monday in May (Memorial Day) and come down the day after the second Monday in October," and separately confirms that "the last day to use the cables is the day after the second Monday in October." In other words, the day the cables "come down for the season" is the final day hikers are permitted to use them, not a day when they have already been taken out. A valid permit dated for that day lets you climb to the summit via the cables before the park removes them after the season closes. Source: https://www.nps.gov/yose/planyourvisit/halfdome.htm Source: https://www.nps.gov/yose/planyourvisit/hdpermits.htm

These dates are not fixed. Both NPS pages note that the cable schedule is "subject to change based on conditions," so the actual up and down dates can shift earlier or later depending on snow, weather, or trail conditions in a given year. Anyone holding a permit near the start or end of the season should check the park's current conditions page before hiking. Source: https://www.nps.gov/yose/planyourvisit/halfdome.htm

There is also a financial safeguard built into the permit system. The $10 per-person Half Dome fee, charged only once a permit is awarded, is fully refundable if the cables are not up on the date the permit is valid for. This confirms that the park treats a permit for the scheduled last day as a day the cables are expected to be usable, since the refund is tied specifically to cases where the cables turn out not to be up, not to the final scheduled day itself. Source: https://www.nps.gov/yose/planyourvisit/hdpermits.htm

As with any Half Dome hike, conditions on the day still matter. The park warns against ascending the cables when storm clouds are nearby or the rock is wet, since slick granite on the cables is the leading cause of fatal falls, regardless of what calendar date it is. Source: https://www.nps.gov/yose/planyourvisit/halfdome.htm