A triple bond is a sigma bond aligned between the two atoms and two pi bonds. The pi bonds are each formed by p-orbitals perpendicular to the sigma bond on each atom that overlap with each other.
A quadruple bond would be a single bond and three pi bonds. There are only two directions perpendicular to the sigma bond. To get a third pi bond pond you would need a third direction perpendicular to the sigma bond. You would need a fourth spatial dimension.
*This is a simplified explanation without getting into quantum stuff.*
Electrons are particles with the same negative charge, so they repel each other.
They are attracted to positively-charged atomic nuclei. If two atoms are nearby electrons try to position themselves inbetween, attracted to both nuclei at once and forming a bond.
Up to a triple bond the attraction of electrons to nuclei outweighs the electrons’ repelling forces. If you try adding more, the repelling force gets too strong and “extras” are thrown out.
Bigger atoms with d-orbitals may form quadruple bonds because their outer, bonding electrons are further away from each other.
The bonding locations are not freely movable, just somewhat “stretchable”. Those of carbon point in 4 different directions best seen in [methane](https://d2cbg94ubxgsnp.cloudfront.net/Pictures/480xAny/9/5/7/140957_Methane-3D-balls.png). To use all four bonds would require to “bend and stretch” at least one of them a lot. Three still works as seen with [molecular nitrogen](https://atlas-content-cdn.pixelsquid.com/stock-images/nitrogen-molecule-4oOwY9D-600.jpg).
To put it very simply, electrons repel each other so they don’t like to be together in a large group. A triple bond has 3 pairs of electrons “close together” and that’s the most they’re willing to squeeze.
A quadruple bond will need 4 pairs of electrons “close together” and they go “nah that’s too much man I ain’t squeezing with so many others”.
As for the reason why the threshold is 3, u/Chromotron gave a bit more elaboration.
Latest Answers