Storage And Toning.

It would be a unique listing "Toning Case"! Keep it in the garage until you decide what to do! I think you were sold a lemon. I will keep an eye on my coins just in case.

It sounds to me like your problem is the "velvet"-and-plastic trays; it is probably outgassing something sulfurous. Do the plastic trays smell, well, "plasticy" or chemical-like? Something that reacts with silver that quickly should be detectable by nose. Does regular silver dip remove it? If it doesn't then it's outgassing something worse.

Franklin Mint back in its heyday in the 1970s shipped out their proof sets with plastic pseudo-velvet cases to display the coins in. Anyone who followed the instructions that came with the coins and placed their coins into the velvet-lined box unwittingly destroyed their coins, because the pseudo-velvet was made out of a plastic which outgassed something acidic. After a decade sitting under the couch, ugly toning on the silver coins was the least of the problems; the base-metal coins simply corroded away.

Trays directly to the bin. Perhaps wash the case out with something like Isopropyl alcohol and leave it open for a time. Get new trays, but put a coin in the case on it's own to make sure that whatever it was has gone. Then use the new trays.

I am very much inclined to agree with Sap. An acidic sulphurous environment could well be the problem.

The glue holding the velvet in place is out gassing, or there is some sort of pollutant (aggressive dust?) within the velvet.

I do not think the aluminum is the cause of the problem; the metal in this case should be preferentially affected before the coins should be.
That is despite the fact that in most cases, aluminum normally and very quickly develops a protective oxide layer over itself. That is why aluminum pots and pans survive.

I kept some silver in paper holders and had them in a cigar box for about 7 years, they got the most beautiful rainbow tone. I'm stuck in bed for the next few days at least (been here since friday) or I'd go get a photo. One's a Sulla, Venus Victrix obv. denarius, came out gorgeous.