How to tell when an LM334 is limiting the current?
At 25C, the LM334 has a minimal drop-out voltage of about 0.9V between V+ and V-. If the V+ to V- drop is larger than that, the LM334 is in current regulation. Othewise, the load is getting the maximum voltage that the source can provide, and is not running at full current.

The simplest way to detect that would be by biasing a PNP transistor with a scaled-down version of the drop out voltage. When the drop out is at 1V below V+, the base would get, say, 0.7V, and the transistor would start conducting.







simulate this circuit &ndash; Schematic created using CircuitLab

This isn't superbly accurate, but might work well enough. The current through the 25kOhm R1/R2 divider is 40uA/V, and that's an error source. The divider produces about 0.72V at 0.9V drop-out. You can increase the resistance of the divider if you can use a larger load resistor R3 or a higher-gain transistor. The ILIM output is logic-compatible as long as V+ is not too high. Otherwise, you'll need a series resistor between ILIM and your MCU input.

You can also simply use a high- or low-side current monitor/comparator. E.g. if you can waste about a volt of headroom and your load can be floating, you can use LTC1540 with a 1.18k sense resistor, you'll get logic output straight out of it.