Something to think about:
Components in a serial connect must have the same current flowing through them. There is no other path to take. So putting 3 diodes or 3 resistors or whatever in series can't improve the current rating over a single component, as they will all see the same current.
If components are in parallel, they will all have the same voltage across them. So putting components in parallel can't change the voltage rating.
Beyond that, depending on components, there will be complications as Nigel said. Let's consider a simple case. Three 10k resistors in parallel. They will have the same voltage across them, and each should carry ⅓ of the current, right? It's not quite that simple, as 10k resistors aren't exactly 10k ohms. The one with the lowest resistance is going to carry more than its share of the current. But that's going to cause it to get hotter, and increase in resistance.... eventually, they'll reach equilibrium.
With diodes, the situation is much more complicated.