To begin with; What constitutes a "bad" diet?
Poor choices of foods.
If it is merely that you don't approve of someone's food choices, that is entirely irrelevant. Those choices fit together into dietary habits that either provide (a) an insufficient amount, (b) an appropriate amount, or (c) an excessive amount of energy relative to their needs as an individual.
Provided that energy provision is an appropriate amount, they can expect to see results from training regardless of anyone's opinion on how "bad" their choices of foods are. Now, ideally we should be including some healthful choices to ensure that we meet our nutrient requirements as well as just energy requirements... with something resembling the "5 + 2" recommendation for vegetables and fruit respectively, but failing to do so would not preclude results from training consistently while meeting appropriate energy and macronutrient provisions.
Excessive total energy intake.
Here's the thing. It's quite plausible that people could show up and train, but have inappropriate eating habits resulting in an excessive total energy intake. As a result, they would not see fat loss.
However, the trope "you can't out train a bad diet" suggests that they could not address this imbalance by increasing their level of activity. That's nonsense. In actual fact, we can expect best results when activity levels AND energy intake are both high. Think of the outrageous diets some famous athletes are known to have... clearly they have "out trained" what for most of us would be terribly excessive.
It's always about context.
What's excessive for an inactive person might be quite appropriate for someone more active. What was excessive when you were inactive might be quite appropriate now that you are more active, in which case you would indeed have "out trained a bad diet".
However, and this is very important;
Your activity levels must be productive and sustainable. We must not fall into the trap of thinking "I'm not seeing great progress, so I'll do twice as much, burn twice as many calories, and then twice as many again until I see results". At a certain point... no matter how enthusiastic or determined or committed or whatever else you feel on day one of this new regime... at a certain point you are going to run out of steam. What you absolutely do not want is to train your body to require that amount of activity just to maintain whatever condition you're in when that happens.
What you want is a serious but sustainable training and activity schedule, and eating habits that meet but do not exceed your requirements. Those requirements may vary subject to your seasonal goals, or they may just fit a vague description of "adequate but not excessive on average".
The contentious exception.
Athletic condition comes out of athletic performance, when a suitable amount of resources are available to facilitate recovery from & adaptation to training. That means energy, and protein. You need energy to perform at training and at sports, energy to recover from training & sports, and protein available to support an increase in lean muscle mass as an adaptation to training.
As discussed, best results will come when the level of activity is high, and level of fueling is also high. This refers to both amount and quality of activity, but we must not always assume that the answer to any lack of continued progress is always to increase activity levels, especially to levels that we can't reasonably expect to sustain indefinitely.
Other than sustainability, there's another reason we shouldn't always assume an increase in activity levels is the solution to any problem, and that is the one form of "bad diet" that you actually cannot out train: a diet of insufficient total energy provision.
You can't expect great results and improvements in condition while restricting to a less than sufficient calorie limit & total energy provision. You will not perform at your best, and you will not have resources available to recover from what you are able to do, or to adapt with the prioritisation of lean mass at the expense of fat mass. Adding more activity when energy intake is insufficient serves only to exacerbate a problem, rather than address and rectify it.
To summarise though:
You can't out train a diet that does not match your energy & macronutrient requirements, but what most people will judge your diet as "bad" for only reflects their own lack of understanding and judgemental self-interest, and is quite irrelevant.
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