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I know the pass without trace spell is strong, but it seems absolutely ridiculous if characters can add +10 to their already high Stealth bonus. For instance, with a Stealth bonus of +8 (which is perfectly achievable for a rogue), you can not physically roll a total below 19 on your Stealth check.

Is the +10 Stealth bonus of pass without trace in addition to the character's existing Stealth check bonus?

Yes, pass without trace adds to the character's existing Stealth check

The pass without trace spell description says:

A veil of shadows and silence radiates from you, masking you and your companions from detection. For the duration, each creature you choose within 30 feet of you (including you) has a +10 bonus to Dexterity (Stealth) checks and can't be tracked except by magical means. A creature that receives this bonus leaves behind no tracks or other traces of its passage.

The spell doesn't say it replaces your existing Stealth check modifier; therefore, it is added on top of it. And as long as two game features don't have the same name, their benefits stack on a target. (In this case, there aren't even two different features or abilities being stacked; it's simply one bonus stacking on top of the default/existing bonus.)

It's not without limitations

Note that the spell description only says it adds a bonus to your Stealth checks and prevents you from being tracked after the fact; it doesn't make you invisible, totally inaudible, unsmellable (as Jeremy Crawford clarified on Twitter), or impossible to detect using other senses.

In addition, the spell description says that it grants those benefits to targets within 30 feet of you. Crawford clarified on Twitter that it's intended to benefit creatures only while they stay within that radius of the caster, though the caster can choose different targets over the duration of the spell.

In other words, the party will still need to stay out of sight - they'll still need to follow the rules for hiding. If the enemy's on the lookout for the party, the party will need something to actually hide behind; if they just stroll out into the open, the enemy will notice them, spell or no spell.

Quick answer: Yes, it does add, however stealth is not an invisibility cloak, but rather a proficiency in utilizing your surroundings through observation and strategy.

Looking at one, if not the best groups that could use stealth, the ninja, we can learn a lot how stealth works. They remembered to both mask themselves from all 5 senses while keeping strategy over strategy to stay one step ahead. Here's a few keys they focused on:

Observe your surroundings

Like V2Blast mentioned, hiding in a dark corner won't mask your sound, breath, or smell, and it won't guarantee that you're not seen either. With enemies knowing darkvision or having beast senses, it's more important to observe not only places to hide, but paths you can take to move, exit strategies to lose a tail, the look of the room (how dark and the pattern/color scheme) as well as keeping in mind the abilities of the enemies you're facing.

They had many tactics for this, from observing hiding places, paths they could take with many exit strategies, and how to avoid detection, or distract from pursuit.

Blend, don't hide.

As the prior section mentioned, hiding isn't the best strategy. With darkvision, observance proficiencies and senses of smell and hearing in play, a bundle of black cloth in a dark brown corner does stick out like a sore thumb. Even combining the spell darkness won't help hide you when you're in a brightly lighted room... The focus is to be proficient in blending with your environment, knowing where you can almost merge with the surroundings and when to leave.

Note: This tactic can also be related to abilities ninjas used like the tanuki no jutsu (or racoon perch) where staying in a high place (above 45 degrees from vision line) is often a place where people don't look. And when you do come down, assuring that it's behind the enemies' back. Wherever you go, Stealth is the mastery of knowing how to find your path from point A to point B with minimalizing chances to hide when possible or creating opportunities to do so when needed.

Know your advantages

Stealth is, in a way, the opposite of you being able to slip past guards on your own. As mentioned before, it's an ability to know how to use your environment, enemy behaviors, etc. to your advantage. That said, your team can also create an opening by having a mage create a sound down a hall to attack their eyes away from you as you whip by like the wind, or know how to utilize both darkness and light to your advantage. For instance, say you were trying to sneak past a guard and the only hiding place you had to hide was in front of a large bonfire. Rather than thinking you're screwed, high stealth could allow you to successfully cast the light cantrip to blend into the flames until they look away.

You might also be the one to use stealth, but note that this ability will increase if your team helps you out.

In short: High stealth will allow you to sneak through rooms unnoticed, but not because you're invisible. Rather, it is a reflection on how well you mastered the observation of paths, the timing of your travel, and disarming of potential obstacles that make up the journey as you sneak. For that reason, having a guarantee in this proficiency isn't ridiculous, but achievable through practice and training.