These fuses are not precision devices. The nominal ratings are at room temperature and reflect the difference between the current the device is guaranteed to carry vs. the current it is guaranteed to open at (after some unknown delay or specified delay). You generally cannot actually use the device near the lower rating, nor can you assume it will open at the higher current within a reasonable length of time. They're just rough guides.
To determine how these things behave you need to refer to manufacturer's data. In the case of the RUEF300, nominally a 3A device made by Littelfuse we have this specification sheet and this catalog/datasheet. Other devices using the same principle will behave similarly, so take this as a representative sample.
From the first one we have this snippet:
Our nominally "3A" fuse is guaranteed to carry 3A @20'C and guaranteed to open within 10.8 seconds when carrying 15A. That's a 5:1 range.
The 6A trip current rating here has no time specified, but it is certainly more than 10.8 seconds that the 15A applies to.
But we have not even taken temperature into account yet.
So if our board gets hot, we may not be able to count on our "3A" fuse to carry more than about 1.5A.
The current to cause trip within a fixed time (10.8 seconds) will go up from the maximum 15A when the temperature is lower than 25'C.
There are other imperfect aspects- the interrupting current is 100A DC/70A AC, so it may not open properly if the fault current exceeds that, and the maximum voltage is 30V, so the open voltage should not be allowed to exceed that rating.
They are basically thermal devices so they require a significant voltage drop to open, which may affect your circuit.
Finally, being thermal devices, the mounting can affect the trip current. It's more important with SMT devices, but if you were to, say, pot the through-hole device described above, the trip current would increase.
As @MichaelKaras points out in the comment, the devices "wear" and characteristics change every time the device trips. The specifications are for a new device. From the IEEE paper Failure Precursors for Polymer Resettable Fuses the below graph shows the change over operations for some typical samples. Resistance also tends to increase with number of operations.