Comment on any of these if more detail is anted:
Easy enough electronically, and I'd recommend an electronic solution - but a relay with at least either 1 x changeover contact or 1x make and 1 x break contact will do this. Relay is rated at lowest value current that you want. Place a wirewound pot of suitable rating in parallel to desensitise it by shunting some of the current.
The relay's break contact disconnects the supply. The relay's make contact self latches itself.
Electronic: A foldback current limiter will do what you want. When current exceeds a trip point the circuit adds resistance until some lesser value is reached. This can be latching.
Example from here - at bottom of page. Circuit explanation given on that page.
Measured and calculated performance (from their site).
A circuit that powers itself completely off when a set current is exceeded is also "easy enough".
eg Placing a diode in series with R4 in the above circuit so the diode conducts when LM317 output is low, and lowering the value of R4 to around a few hundred ohms would probably make the circuit fully latching. May not then start well - may need a little thought to tidy the result.
Here is an electronic fuse circuit using a relay as the output switch and a flip flop to latch the tripped state. The relay could be replaced by a MOSFET and the latch could be replaced by hysteresis feedback.
I drew up a circuit before I saw the above which does much the same but removes the latch and uses a MOSFET. It uses 1 x opamp section, 1 x MOSFET, 1 a voltage reference (or a zener with less precision), 5 or 6 resistors, a diode and a pot to set trip level. Turnoff is complete and instant on exceeding Itrip and it latches off. Trip delay can be added with one capacitor and operation can be switched to constant current limit by opening the diode circuit. Can draw (slightly) more tidily and post if needed.
Here is their circuit.
3 transistor foldback current limiter:
Second circuit on page here.
This too can be made to latch off.
A few idea starters here Gargoyle "foldback current limit circuit" image search
Gargoyle "electronic fuse circuit" image search
Just using a constant current source may suffice - so that current can not go above some preset limit.
Fuses don't protect light bulbs or lamps - they can do this themselves by burning out and then you replace them. In the main, a fuse is there to prevent a fire and this means it protects the wiring from over-heating and melting the insulation and thus causing a fire.
So, rate your fuse accordingly to protect the wire. If your wire is rated continuously for 10 amps then use a 10 A fuse and it doesn't need to be a quick blow either - anything a little bit over 10 amps might take several minutes to blow but the wire won't have got that hot in that time. 20 amps would probably take the fuse out in about 1 second. Here's an example of a 30 amp fuse: -
And here's another guide that shows the process of selecting the right size cable and providing an adequate fuse (taken from this very good page): -
Best Answer
No.
Both current and voltage rating are the "safe operating conditions" for the part and your application should stay below these limits. Above this limits, the behavior is "undefined". Chances are, that the fuse will just pop at some point, but that's not guaranteed and you shouldn't design that way. If you need over-current protection, put in an extra current-triggered fuse.