I've played Steam rather than Age of Steam, but I think it's similar? In any case, I found this clarification on an online Age of Steam FAQ:
Q4. As mentioned in the section titled "Moving Goods", a player may
forego shipment during a round in order to "increment" his engine link
level. It does not specify in the rules by how many links a player may
increment his engine, only that the maximum is 6 links. Can a player
increment his links by more than one per round? In other words, can I
go straight from a 2 link engine directly to a 6 link?
A4. You can only move your engine level up by 1 for foregoing 1 cube
shipment. You are only allowed to do this once per goods movement
phase. Note that the word "increment" means "To increase the amount of
ONE unit". Also be aware that if you get the Engineer you can still
forego a shipment in order to get a maximum upgrade of +2 on the turn
as a whole.
I can't vouch for the accuracy of this ruling, but it does tally with what I seem to remember from playing Steam, that you couldn't just increase your Engine madly early on, it was a slow, gradual process. So I'd say once per goods movement phase sounds like a pretty safe bet.