Gabriel Barkay is probably the world’s leading scholar on tombs in Jerusalem.  He is quoted in a story by the Agence France-Presse.

Gabi Barkai, a professor of archaeology at Tel Aviv’s Bar Ilan University who has 40 years of experience excavating tombs in Jerusalem, confirmed the nails dated back to the first century, but said it was impossible get a more accurate date.
“Nails are a rare things in tombs from Second Temple period Jerusalem,” he told reporters, noting that there was no bone residue attached to them.
There were several theories as to why they might have been put inside a grave, one of which was that crucifixion nails were believed to be powerful amulets for the afterlife.
But there was “no proof whatsoever that these nails came from the cave of Caiaphas,” he said.
Asked if he believed they were used in the crucifixion, Barkai was cautious.
“It’s a possibility,” he said.

I take the last quote in the sense of, “It’s not absolutely impossible that these nails were used in a crucifixion.”  It should also be noted that Barkay was likely a paid consultant for his appearance in the movie, and he would have been reluctant to completely dismiss the movie’s basic premise at a news conference sponsored by the film producer.

Time magazine has this:

Also unclear: Why would a priest be buried with a nail? Jacobovici points to scholarship indicating crucifixion nails were regarded by contemporary Jews as holding special healing powers. The bit of paganism was apparently tolerated, even in priestly circles: a woman’s skull found in the same tomb contained a Roman coin, presumably included to pay the boatman steering souls across the River Styx.
Gaby Barkay, a professor at Bar Ilan University and probably the most prominent archeologist in Israel, offers another explanation. Jews at the time of Christ “were impurity freaks,” Barkay says. Anything in the vicinity of a corpse was thought to be contaminated by death, even a nail stuck in a nearby wall. “Therefore it would probably be removed and put into the grave,” he says.
The professor quibbles with other assumptions as well, but notes that “nails in general are a rare thing in tombs of the Second Temple Period,” and his presence at a crowded news conference has added weight to Jacobovici’s effort.

Barkay may be one of the most credible archaeologists in Israel, but he’s certainly not the most prominent.  While it is true that crucifixion nails were considered sacred amulets in the ancient world, that is usually the explanation given for why nails are not found in tombs.

The Christian Post adds:

According to the documentary’s guest archaeologist, Gaby Barkay, iron nails were rarely found in tombs and were normally used to carve names in the stone ossuaries.
“There’s no proof that the nails are connected to any bones or proof from textual data that Caiaphas had the nails for the crucifixion with him after the crucifixion took place and after Jesus was taken down from the cross,” Barkay said. “On the other hand, those are possible things.”

Everything is possible.  The question is, what is likely given the evidence?  Should profit motives influence our evaluation of a claim?