The thing you probably haven't learned is that particles appear differently depending on what distance you observe them from. In particular, the charge and mass of electrons are so called “running constants” and depend on the distance. The thing is: the difference is very-very small on the human scale. You actually can't measure any difference on the scale from one millimeter to few thouthand kilometers. To see something, you have to significantly zoom out (like when observing very far galaxies) or in (like in Large Hadron Collider).
It can even happen that on infinite distances some parameters (like electron mass) become zero, and on zero distance some other parameters (like electron charge) become zero as well. It's not a problem, because the relevant physics always happens inbetween. It's actually even a problem solver. Recall, two charges of the same sign repell each other proportionally to inverse square of their distance. On the distance zero the repulsion becomes infinite. Any electron being a point partice should classically have a huge problem with this fact. It doesn't; on zero distance, it has no effective charge.
Interesting problem is though, that in some theories there is a distance beyond which we cannot scale, a zoom limit also known as Landau pole. At this distance some of the properties become infinite. This actually happens to all known quantum field theories if you consider them in context of unmodified Einstein gravity. This directly hints to the fact, that these theories cannot be ultimate. They just describe how the ultimate theory appears when viewed from sufficient distances. The ultimate theory could be for a string theory, a quantum field theory on a nonclassical background (noncommutative spacetime, spin network etc.) or a usual quantum field theory with substantially unusual gravitation. Or something else we have so far no idea of.