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Death Knell is a spell that among other things grant a temporary +1 increase to caster level.

I have seen different answers and questions here that seem to assume this spell stacks with itself.

Are they right or wrong?

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No, it doesn't stack

This has been answered in a FAQ:

Temporary Hit Points: Do temporary hit point from the same source stack?

No. Generally, effects do not stack if they are from the same source (Core Rulebook page 208, Combining Magical Effects). Although temporary hit points are not a "bonus," the principle still applies.

This prevents a creature with energy drain (which grants the creature 5 temporary hit points when used) from draining an entire village of 100 people in order to gain 500 temporary hit points before the PCs arrive to fight it.

Temporary hit points from different sources (such as an aid spell, a use of energy drain, and a vampiric touch spell) still stack with each other.

—Pathfinder Design Team, September 2013

This includes the spell Death Knell, spell-like abilities that cast Death Knell, wands of Death Knell, and magic items that have the effects of Death Knell.

Although the FAQ is about temporary hit points, the principle is that bonus from the same source should not stack, as described on the Core Rulebook, under Combining Magical Effects.

Spells that provide bonuses or penalties on attack rolls, damage rolls, saving throws, and other attributes usually do not stack with themselves. More generally, two bonuses of the same type don't stack even if they come from different spells (or from effects other than spells; see Bonus Types, above).

Even when an effect is not labelled as a "bonus", if the effect increases it, it is gaining a bonus. Just like the spell grants you bonus temporary hit points (normally you have 0).

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    \$\begingroup\$ I'd like to highlight, since one of the links mentions that the increase to caster level isn't worded to be a bonus, the sentence "Although temporary hit points are not a "bonus," the principle still applies." in that quote. Increase in caster level may not be a "bonus", but the principle should still apply. \$\endgroup\$ – user36958 Nov 13 '17 at 12:38
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    \$\begingroup\$ @SimpleLime the way the spell is worded, it is a bonus. "your effective caster level goes up by +1" reads as "you gain an untyped bonus of +1 to your caster level". The confusion is normally on temporary hit points, that are worded as "You gain X temporary hit points". \$\endgroup\$ – ShadowKras Nov 13 '17 at 12:42
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Sort of

Temporary hp and bonuses don't

Temporary Hit Points, as this answer notes, have been FAQ-errataed to be treated as if they were a bonus for the purpose of self-stacking, and from that the FAQ draws the conclusion that they no longer self-stack. Thus, the rule:

Bonuses without a type always stack, unless they are from the same source.

is interpreted as meaning to imply that bonuses and temporary hp from spells never stack with themselves unless the spell specifically says so, even though in actuality those rules merely state that untyped bonuses behave thusly (and the rules for typed bonuses mean that most other types of bonuses work the same way as well, but notably not dodge bonuses)

Non-bonus benefits do

Death Knell's increase to caster level, notably, is not written as a bonus. Anything that's not a bonus is not affected by the rule that untyped bonuses don't stack with themselves. The reasoning in the FAQ may lead us to believe that designers intend no beneficial effects of spells to self-stack, specifically because that situation can lead to what they consider problematic village-slaughtering outcomes, but just as the rules clearly do not prevent temporary hp from self-stacking without that FAQ, they also do not prevent not-bonus increases from self-stacking.

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