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If the following two items are both used, do the bonuses stack?
Ring of jumping, improved : +10 Competence to all Acrobatics made to make high or long jumps
Boots of Striding and Springing: +10' Enhancement bonus to base land spd; +5 Competence to Acrobatics to jump
Would that net a +15 to jump checks? They are from different sources, so I would think they would stack.
If they are both Competence bonuses, no, they would not stack. That they are different sources makes no difference. The 'no stacking' rule applies with any bonus that is the same, different item (source) or not, with the exception of a dodge bonus.
They don't stack. However, note that the +10 bonus to base land speed also grants you a +4 bonus to jumping (provided your land speed is at least 40 after equipping the boots).
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A competence bonus (or penalty) affects a character's performance of a particular task, as in the case of the bardic ability to inspire competence. Such a bonus may apply on attack rolls, saving throws, skill checks, caster level checks, or any other checks to which a bonus relating to level or skill ranks would normally apply. It does not apply on ability checks, damage rolls, initiative checks, or other rolls that aren't related to a character's level or skill ranks. Multiple competence bonuses don't stack; only the highest bonus applies.
Nope.
But a potion of jump does stack with a ring of jumping.