an LM317 Constant current is different than over current protection. if you setup the regulator in constant current mode, you can regulate up to 1.5amps through the regulator, but this configuration ignores regulating voltage.
if you setup the lm317 as a voltage regulator, you cannot regulate the current going though it with just that one device. they do however have internal over current protection that will turn off the regulator for like a second or so if they see more than 1.5(or possibly up to 2.2 as grumpydoc stated) they will shutdown, similar to how they shutdown if they get too hot. so you cannot use this feature as a way to regulate current, only as a way to protect your circuit from over-current conditions, like a short. if you want to regulate both voltage and current, you would need to either use 2 regulators, one configured for voltage regulation, one configured for current regulation, or you would need to design your own external current regulation, similar to how Dave did in his power supply design video.