Voltage and current tolerances
Staying closer, perhaps, to your Question
It's all too easy to wind up discussing things at cross-purposes: two different people imagining two different things but where they both incorrectly believe the other imagines what they do. Sometimes, the results can be quite humorous. At some risk, I'll try to help disambiguate a few ideas that I merely imagine are helpful.
The phrase &quot;current-tolerance&quot; suggests a few things at once to me:

A device that acts as a current source or sink of a specific magnitude, but where there is also a specified &quot;tolerance&quot; around that specific magnitude that a consumer can expect as normal for the device and would not be considered incorrect operation if it varied by that much from the specific magnitude over the range of operating conditions also specified (like temperature, humidity, compliance voltage range, phase of moon, etc.) and/or prescribed test conditions.
The limiting magnitude of current over similarly prescribed test conditions of a device. Usually, this is some kind of conductor or conducting switch that acts as a &quot;short-circuit&quot; when activated. All wires and switches have some kind of limiting magnitude over which they are expected to work (taking into account all of the conditions they also specify for the device.) Switches have to take into account a lot of specific details with respect to making and breaking contact, as well as conducting properly when active. Wires not only have to take into account proper operation without melting or sagging or stretching or otherwise failing their purposes over extended times, but also have to take into account other details such as the impact of heating on their insulation materials (if present), fire safety issues, voltage drop vs intended use of the conductor, and so on.

Note that in one case it's a \$\pm\$ kind of thing, as in \$1\:\text{A}\pm 10\:\text{mA}\$, and in another case it is an absolute maximum, as in \$\le 5\:\text{A}\$.
I could add more but we've already listed current sources, switches, and wires and that's enough to make the point that the phrase &quot;current-tolerance&quot; cannot be expected to communicate well if you also do not provide the context, too. It's too broad of a phrase and therefore doesn't have a &quot;bright line&quot; meaning. Different specialists, lacking any other context, may apply any number of meanings to it. And if you don't know their specialty well, you probably won't know how they are taking it -- or if they take it well, at all.
The phrase &quot;voltage-tolerance&quot; also doesn't have a bright line meaning. I'll copy the above pair and modify them to account for a few things, again:

A device that acts as a voltage source of a specific magnitude, but where there is also a specified &quot;tolerance&quot; around that specific magnitude that a consumer can expect as normal for the device and would not be considered incorrect operation if it varied by that much from the specific magnitude over the range of operating conditions also specified (like temperature, humidity, compliance voltage range, phase of moon, etc.) and/or prescribed test conditions.
The limiting magnitude of voltage over similarly prescribed test conditions of a device. This might be for a insulator on a specific wire product where they cannot guarantee proper operation if the wire inside the insulator is subjected to differential voltage across its thickness that exceeds some &quot;volt-per-meter&quot; stress they are promising for the insulating jacket. Since they know the thickness, they can work out the maximum guaranteed voltage difference and provide it as a specification. This could just as well apply to capacitors, which necessarily use an insulating material as their dielectric, too. But again it could apply to switching devices when they are not actively engaged, etc.

Note that once again in one case it's a \$\pm\$ kind of thing, as in \$1\:\text{V}\pm 10\:\text{mV}\$, and in another case it is an absolute maximum, as in \$\le 5\:\text{V}\$. And not also once again that lacking any other context, there may be any number of meanings.
Wanderings
As far as:

... understand what's happening a little better on the physical side ...

Hopefully, the above allows you to realize that trying to understand the physical side of something has a lot more to do with knowing as much of the context as possible. That context doesn't just include a complete drawing, for example -- by this I mean to imply all of the details of its manufacture -- but it also includes the all of the circumstances within which it is placed.
Underlying the very idea of a device is that there is a physicality you can mentally identify as something you might hold in your hands or otherwise touch. But also underlying that same idea is all of the rest of the universe that is not the device, but surrounds it. It's really the implied boundary that you are placing around something that has things it includes and things it excludes. So when I say &quot;this diode&quot; I mean the indicated diode and its internals but I also mean everything around the outside of it which is not &quot;this diode.&quot; You cannot say &quot;this diode&quot; without meaning all of that.
The way &quot;this device&quot; physically behaves is just as much about what isn't &quot;this device&quot; (it's surroundings that it is locally interacting with -- and even then I'm assuming all interaction is local and we already believe even that isn't entirely true -- just mostly so) as it is what is &quot;this device.&quot; A thermometer is a bit of matter that is inserted into another bit of matter and where these bits are allowed some time to interact with each other before taking a reading. To understand the physics of a thermometer and why it works at all, you have to understand a great deal that includes conservation laws of physics that operate at the atomic/particle level, large number population statistics and probability, details of the thermometer and what is &quot;being read&quot; as a measurement itself, as well as emergent phenomena that do not exist -- even conceptually -- at the atomic/particle level but do arise when you are talking about unimaginably large numbers of possible initial states, all possiple permutations of them, and the likelihoods of specific measurements that hypothetically could result as a measurement, some moment in time later.
So, when you want to understand something, you also need to specify &quot;at what level,&quot; too. Each specialty or interest will have a dominant level of understanding and other levels about which that specialty or interest will &quot;simplify&quot; as a practical matter, as such details don't impact them (much.) A very good example of this is that current in a wire is caused by very very slight and gradually changing concentrations of electric charges at the surface of the wire along its length. Assuming the wire is unvarying over its length, these electric charges at the surface will be neutral at the half-way point along its length. But they will not be neutral away from that central point in either direction. Very very few electronics designers worry about this detail. Yet this detail is the physical reason why a current occurs, at all. And that is something that every electronics designer cares about. So, while they care about something important to them -- current -- they do NOT care about WHY it happens. They just know it does and they have other ways of working out the current without having to worry about things like &quot;static charge gradiations over the length of a wire&quot; when they analyze current.
So if you were to prod me to help you

... understand what's happening a little better on the physical side ...

with regard to current in a wire, I would be mystified. Were you asking about these static charge gradiations? Most certainly, that would help you understand things a little better on the physical side. But you really may have no clues about that and are really asking for a much less complex answer. (And there are still deeper levels about which you can very easily dive into, which elaborate still further -- as gradiations are themselves just a simplification of still deeper behaviors.)
This is why it is so difficult to answer questions directing an answer towards the &quot;physical side&quot; of things. Reality is very, very complex. We understand it by reducing the complexity into &quot;lumped&quot; ideas that ignore a great deal of reality. Usually, we prefer to move away from the physical side, as we could spend our entire lives on that topic and never fully grasp it. It's a bottomless pit, so to speak. (Turtles all the way down?)
The fact is we don't even know for certain that atoms exist. All of science knowledge is based upon certain assumptions (axioms) that span a wide spectrum from mathematics to logic to certain assumptions we feel are &quot;reasonable&quot; to make about reality itself. If any of those axioms are shown to be false, everything built upon them may change as a result. Science knowledge is also based upon the correct application of those axioms. If some mistakes are made there and later uncovered, again things can change. Finally, science knowledge does not have a crystal ball and has no idea what the future may yet bring. That may cause those assumptions or the soundly reasoned application of them to alter under the weight of new experimental result. So all science knowledge is ever-tentative, open to being changed in small or large ways. There is no &quot;absolutely true forever&quot; in science knowledge. Only &quot;very very strong theoretical and experimental evidence, so far&quot; that weighs very very heavily in this particular way and where consensus currently tells us that it is highly unlikely that the future will bring us something new to change it. How much of a demarcation is present there, itself depends upon just how well explored something is.
Science, as it has evolved these last few centuries, is in many ways a radically new way of thinking about the world around us. It's not a natural way of thinking, either. It mistrusts what our senses tell us in fundamental ways. And it mistrusts all conclusions we make, instead preferring theory and the results of good experimental design -- the way we approach the world around us -- over the conclusions we make about the world.
The smartest people in science all were and are wrong about most everything they believe. What matters aren't their conclusions (Einstein's earlier idea that the universe was static and neither expanding nor contracting.) What matters is how they approach gaining some kind of new understanding of it. We read from them, not for what they say is true, but for their brilliance in walking a path in an attempt to understand. How they think, not what they think.
In that vein, I highly recommend spending some time reading Galileo's &quot;The Assayer,&quot; &quot;Dialogues Concerning the Two Chief World Systems,&quot; and &quot;Discourses and Mathematical Demonstrations Relating to Two New Sciences&quot; (&quot;Discourses Concerning Two New Sciences.&quot;) Wrong about most everything, but absolutely brilliant in how he investigates. For example, he was able to mentally demonstrate that there had to be some kind of &quot;glue&quot; between the smallest particles (he knew nothing about atoms, but assumed some &quot;smallest particle&quot;) of copper through a simple examination of observables. We know these as &quot;bonds&quot; (another of those simplifications of something still deeper) today. It's stunning what he was able to unearth from detailed examinations and careful, crafted thinking about them. One can learn a great deal about how to think by reading those three pieces of work.