Low value manually resettable adjustable electronic fuse
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.