A diode starts conducting with a few fA. The current through the "resistance" of the diode causes a voltdrop. The higher the current, the higher the voltdrop. But this "resistance" is not a constant like a normal resistance. It follows an exponential curve. So if the meter sources 0,4 mA the voltdrop will be 0,551 for your diode, at 1 mA it is 0,6V. With 10 mA it wil be higher, lets say 0.7V and at 1A it can be over 1 V.
At first there is a big increase in voltage with a little increase in current, but the rate that voltage increases, decreases the more the current increases. So lets say it is 1V at 1A, then it can be 1.1V at 2A and 1.15 at 5A and 1.17 at 10A (just made up numbers to show what happens, In real life a 1n4148 reacts different, that will be some smoke at 1A, a tiny bit more smoke at 2 A and no smoke at 10A (because there is nothing left to smoke
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