Which level of sloppiness is forgivable at PhD thesis completion The question can be really irritating for most of the people and I am sorry for that. Surely, the right answer should be "no inaccuracies, everything should be right!". However, let's take a look at real situation...

...when you realize that you were sloppy PhD student and carrried some experiments in inaccurate way, but there is no time to repeat them again.
You try to explain it in your thesis that the observed effects are still there, just the exact quantification is not clear because of raised issues.

Is it possible still "defend" thesis? You learned from your mistakes how you should not do it in the future and PhD thesis is a qualification work, so...are mistakes forgivable at this position of PhD student?
How to formulate an answer for the defense, which reflects that you understood it was not right what you did and you will do better next time?
As you describe it, I'd say that it isn't "forgivable" at all. 

But, it may be that you sneak through the process and get the degree. 

Doctoral research is about extending knowledge, just as is all research. The difference in doctoral research is that you get a certain kind of help and guidance in it and a certain kind of evaluation of your results. Your results don't need to be earth-shattering, but they need to be correct and to extend knowledge in some way. 

But in mathematics, if you can't prove the main result, you really don't have anything. I chemistry, if your technique is so sloppy that your results aren't valid, you don't have anything. In biology, if you make certain kinds of sloppy errors but your results are believed, you can kill people. 

I question your example, in fact. If your experiments were inaccurate and they show "the observed effect" how do you know it isn't just experimental noise? Every experiment results in something, of course. 

But experiments can fail in a lot of ways, just as can mathematical proofs. Sometimes, it is possible that one learns something unexpected from carrying out an experiment and the result gives knowledge that points in some new and valuable direction. The same is true of a failed proof in math. It may give insight into some other ideas that are valuable on their own. But that is very unlikely to result from sloppiness, as the sloppiness itself makes analysis very difficult. 

I would guess that trying to finesse it with a committee is likely to fail. Someone will probably object if you just "promise to do better in the future". There is no evidence of that in your past. Passing you would reflect badly on them and on the university. Only luck and the carelessness of the committee will save you in such a situation. 

But I'm pretty sure it happens. Rarely, I hope.