skill bonuses from armor unlock the next tiers?

[AlteredDominant]

Do the bonuses from armor unlock the next skill tiers?  If so, how do I confirm that in game?

My sneak skill is at 35 base.  I equipped a +5 to stealth skills armor, hoping to bump me up to 40 for pickpocketing, but it doesn't work.  My adjusted sneak skill shows (40), but the pickpocket interaction doesn't appear.  However, if I put points into sneak up to 40, it does appear.  

So, now I'm curious how adjusted skill values actually work.  If they cause me to cross over into the next tier, how do I confirm they're working?   It seemed to have worked with the persuade skill.  I had under 40, equipped a +5 item and then passed the check.   

[Anhur]

The equipment stat bonuses simply bump up that stat. Let's say you find a lock that you need 55 lock picking to open but you only have 50, you put on a piece of armor that has +5 to locking picking and now you can open it. The skill abilities are connected to your actual skill number.