I will offer a contrarian perspective here, and propose that neck-deep functions differently from knee-deep often enough to warrant a clear distinction, although this by no means lets knee-deep off the hook as being necessarily "not that bad." See penultimate paragraph and corresponding example for caveat.
I notice in newspapers around the period from 1910 to 1920 that the phrase "neck deep" is used often in a literal sense to refer to people that are, for all intents and purposes, completely submerged and only floating in the natural position for a human being in water, which is exactly neck deep.
When figuratively extended, this is almost never used to describe a good thing.
Germany is neck deep in Russia fighting
writes the Chicago Tribune (paywall) in 1918.
It seems to me that the figurative expression, like the literal one, might carry more of a connotation of being stuck in something to the extent that your feet are not on the ground, as in the case of Germany warring with Russia, whereas knee-deep actually draws attention to the legs and their movement. Knee-deep, then, is well-suited to describe a situation in which one is beleaguered by obstacles and proceeds, whereas neck-deep might suggest that the situation is too dire to proceed.
Consider some case studies:
Kimberly Howard was knee-deep in planning her wedding when she and her then-fiancé heard some friends talking about a service they used to plan their own honeymoon.
One would be less likely to say that they were neck-deep in wedding planning, since the activity is generally a positive and optimistic one, and doesn't imply a complete incapacitation of the person undergoing the planning.
In an article dated today in People magazine, actor and comedian Patton Oswalt relays a story about the death of his wife.
“I was still neck deep in grief and suddenly a single father and not sleeping, but I knew that I wouldn’t be able to live and go on with life if this was left undone,” he told Meyers of finishing I’ll Be Gone in the Dark. “I basically begged [the co-authors] and the publishers to find a way and they were the ones to find a way and finish it.”
Here again, neck-deep seems fitting in describing a kind of grief so crippling that it forms the basis of an existential struggle. One would not be likely to say "knee deep in grief," perhaps because the implication would be likely to strike a listener or reader as not quite bad enough.
My conclusion is that no, the mere existence of "neck deep" as an expression doesn't have any relative implication on uses of "knee deep," but because each expression corresponds to a quite different literal situation, the figurative extensions have somewhat unique connotations, and it could be said that there are some cases where "knee-deep" and "neck-deep" could both be used, and some cases where one would be an unlikely fit compared to the other.