It is not the voltage rating that is the problem. The category rating on meters is the ability of the meter to stand up to transient voltages from such things as surges from lightning, grid switching, large capacitors and so forth.
As I have said a category II or III in most cases will be fine for checking voltages around most residential branch circuits, but the closer you get to the supply source the more likely to run into voltages transients and the higher the category rating of meter will need to be used.
Voltage transients are a very common occurrence in AC utility furnished power.
Checking the amperage of a conductor is just as dangerous if not more so than taking a voltage reading. When taking an amperage reading most people get a false sense of safety due to the fact that they are not coming in contact with a live terminal such as doing a voltage reading.
During an amperage test the meter can be subject to more voltage than doing a voltage reading simply due to the makeup of the meter. The jaw of an amp probe is a simple transformer winding that opens and the closes around the conductor. It is again opened to remove. In both cases bare metal is exposed to the live circuit. Arc flashes have occurred during amp readings.
The fuse internal to a meter is not a rated fuse. Most are simple AGC/SFE type fuses and these will arc over if they get started. The ionization of the air around the fuse will conduct current. This I have witnessed myself. Having a DVM set for an amperage or resistance test and touching the leads to a live circuit can result in a very enlighten experience and with shocking results. Did you catch the pun in that sentence?
Make it a practice to always be checking the setting of you meter to remain safe while using the meter is not bad advice no matter the person making the statement.
Knowing the limitations of a meter is good advice no matter who is giving it.
Understanding the dangers involved with the use and maintenance of electrical circuits is good advice for anyone to listen too. Now you either agree with this or you disagree, which is it?