Why has gold dropped during the same week as Deutches Bank faces great difficulties I chose the phrasing "great difficulties" to be polite. The "capital release unit" seems to be a bit of a desperate action from the bank, coupled with further complications involving bad press made things even worse for DB. 

Wouldn't all of that be a fairly unmistakable signal to buy gold as a hedging instrument?

Question

Is there anything in economic theory that can explain why gold would drop when a bank looks to be in big trouble? Have I missed something? Or is this a case of "market irrationality?" I wonder if there are studies on how long markets can remain "irrational."
As Brian Romanchuk commented, there are a number of factors that can affect the daily price of an asset.  Gold is one of the most traded futures contracts on the CME so the performance of one bank (assuming you mean Deutsche Bank) might not be enough to offset those other factors.  

But there is a strong correlation between gold and fiat currencies.  This is because gold is used to hedge against inflation.  

Regarding your question about economic theory:

In a fractional reserve banking system, the supply of money is controlled by a central bank such as the Federal Reserve.  According to the Quantity Theory of Money, the supply and demand of money determines inflation.  However, John Maynard Keynes rejected this theory because it doesn't account for interest rates. 

Keynes suggested that governments could offset recessionary / inflationary periods by directly decreasing / increasing interest rates.  According to Keynes, lowering interest rates in a recessionary period will create an incentive to borrow (cheaper capital) and short-term demand for money increases.  Prices change slowly in response to the change in interest rates as lenders increase the money supply to meet demand.  This slow change allows money supply and changes in interest rates to encourage or discourage borrowing and lending.

I'll attempt to unite economic theory and the futures market:

Generally speaking, a commercial bank generates most of its revenue through loans and consumer lending.  You could think of it as essentially selling / producing money.  They are inherently long money just like a gold miner is inherently long gold.  So their "product" becomes cheaper when interest rates are cut.  Since the price of gold behaves similarly to currencies, I suppose a bank could short gold futures in order to hedge their position.  

The Fed's chairman Jerome Powell spoke recently. Many interpreted his economic outlook as a dovish signal which could imply lower interest rates.  If that happens, banks' revenue would decrease and the value of the dollar would be expected to decrease and the value of gold would be expected to decrease as well.  But nothing is guaranteed :)  

Regarding irrational markets

A former coworker would say to me, "the market can remain irrational longer than you can remain solvent".  If you are able to predict irrationality then it would no longer be irrational. I recommend you read the book Fooled By Randomness by Nassim Taleb.