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Read somewhere that the cover saves from Jink act like stealth/shroud and thus stack with other, natural cover. Looking over the rule book, I'm not seeing this.

So if I flat out a bike (4+ cover per jink) and go behind some trees (5+ cover) would I get a lower cover save than 4+ or would it just stick with the 4+ from jink?

Both are types of cover that do not stack, so you take the highest as you had suspected.

Cover saves from different sources do not stack. However, stealth/shrouded is a bonus to a cover save, and also specifically mentions that the bonuses do stack.

here are some examples for cover saves regarding an infantry model -

A.) In a Crater (area terrain) = 5+ cover save
B.) In a Crater, but obscured by a ruin wall = 4+ cover save (from the ruins)
C.) In the open, but with Shrouded USR = 5+ Cover Save
D.) In a crater, with Stealth and Shrouded USRs = 2+ cover save (5+ from crater, +3 for stealth and shrouded bonus)