After working in digital electronic, analog electronics can catch many people off guard. A power supply rated up to 1 amp may not work well close to its rated limit. The voltage may drop unexpectedly.
Also, according to this schematic, on the Arduino Nano, between the USB voltage net (VUSB) and the 5 volt Arduino Nano net (+5V), there is a fuse (MSMF050) that operates in the 0.5Amp range and should trip at about 1.0Amps. Pulling the trip current may trigger this fuse.
Some people design with much wider margins. For example, if a device or part requires 1A to operate some people use a power supply who's limit is double or 2A.
Note, there is the fuse in the Arduino Nano that should trip at about 1.0A so using a larger power supply would not work here. Consider supplying power to the external device or part directly from a capable power supply.