We provide a simple model, able to explain why the overnight (ON) rate follows a downward intraday pattern, implicitly creating a positive intraday interest rate. While this normally reflects only some frictions, a liquidity crisis introduces a new component: the chance of an upward jump of the ON rate, which must be compensated by an intraday decline of the ON rate. By analyzing real time data for the e-MID interbank market, we show that the intraday rate has increased from a negligible level to a significant one after the start of the liquidity crisis in August 2007, and even more so since September 2008. The intraday rate is affected by the likelihood of a dry-up of the ON market, proxied by the 3M Euribor—Eonia swap spread. This evidence supports our model and it shows that a liquidity crisis impairs the ability of central banks to curb the market price of intraday liquidity, even by providing free daylight overdrafts. Such results have implications for the efficiency of the money market and of payment systems, as well as for the operational framework of central banks.