Are devices damaged by high current, or high temperature?
First, many devices specify a maximum input current as a characteristic, rather than an operating requrement. For example, if a digital logic IC specifies its maximum input current is 5 mA (for example), that usually means that as long as you provide an appropriate power supply voltage, the device will not draw more than 5 mA --- that is, it is the device's promise not to take more than 5 mA, not a requirement to the user not to provide more than 5 mA.

However, other devices do specify a current limit as a user requirement. Common examples are resistors, inductors, linear regulators. For resistors or linear regulators, the reason for this limit is usually to limit self-heating that would cause thermal damage to the part. For inductors, the limit is more often related to saturation of the core material, although thermal limts can also come into play. In these cases, providing better cooling (say by heatsinking) can allow operating with higher currents, or be required to avoid de-rating the max current specs.


  I'm wondering if we cooled the device at -230 Celsius (we're accepting that it's still working) and then we applied that high amount of current would it survive?


It depends on the type of part. For some parts, it might allow you to operate with higher-than-rated currents. In other parts it might cause thermal stresses (due to thermal gradients and or CTE mismatch between different materials within the part) that damage the part even when operated below the nominal rating.