What does happen when a wire is run close to its maximum "calculated" current?
When a wire is used close to its maximum current it gets hot.

When a wire carries any current it heats up but when used within its design limits it will not get too hot. As the current increases the wire gets hotter. A wire that carries too much current can get hot enough to start a fire. Many cables in houses and other buildings are buried in the walls and so are surrounded by insulation and wood which might not be fire-resistant. The consequences of an overloaded (and very hot) cable in such a wall can be a house fire.

The electrical system should have fuses (breakers) that are set to trip at currents well below the maximum current of the cables. I am sure that the various countries have regulations about the fuses to be used for different cable types in different installations. The goal is that the fuse breaks the circuit, which is a minor inconvenience, long before the house might catch fire.

Given the disclaimer text in you question I would recommend that you choose cables that can carry significantly more than the expected current plus fuses for the expected maximum current. For exact choices I recommend that you talk to a local electrician or to an expert in the shop where you buy the cables and fuses. If your shopkeeper does not know then find a better shop.