Depression, in many cases, is about perception.
Ask yourself what it's like to be depressed. If you're anything like me, you'll use a lot of sense-related verbs. When you have depression, things feel bleak. They seem hopeless. When I actually am depressed, I don't just think that things feel bleak, or seem hopeless. When I'm depressed, things are bleak, and are hopeless.
That's one of the reasons why people who are depressed, in a lot of cases, struggle to believe they have depression. Other people may tell them they have depression, and that what they're feeling isn't true, but it is very hard for a depressed person to believe that, because their feelings exist in direct contradiction of what the other person is telling them. The other person may say things aren't as bleak as the depressed person thinks, but the depressed person feels like they are that bleak.
Imagine trying to convince someone that the apple they are holding in their hands isn't actually an apple, but is actually a tortoise wearing a straw hat. No matter how hard you reason, argue, beg, and assert, the other person is very unlikely to believe that they're holding a hat-wearing tortoise. When they look at what's in their hands, they see an apple. When they touch it, they feel the shape and texture of an apple. When they smell it, they smell an apple. And when they bite it, they taste an apple (as to how the supposed tortoise feels about being bitten, well, best not to think about that).
Now imagine telling the person holding the apple/tortoise that the reason they think it's an apple is because someone has put a mind-control chip in their head causing them to perceive tortoises as apples. Most likely, the person would be somewhere between amused and insulted. How ridiculous is it that someone put a tortoise-to-apple mind-control chip in their head, when it's so clear to them that they're holding an apple in their hand and not a tortoise! People tend to trust their own senses. For the most part that's a good thing, as it was probably the main reason we're not still living in caves. It becomes a problem when the things they're sensing are unnecessarily infringing on their lives.
The best way someone could be convinced that they have that mind-control chip in their head is if a lot of people, whom they trusted, told them that is exactly what is happening. Loved ones and medical professionals would have to line up and explain to them that a mind-control chip is in their head. If the person's still doubting, they might ask questions like when the chip was put into their head, how it works, and if they can't get good enough answers to that, it's all the easier for them to doubt.
And even after telling them all that, and giving convincing reasons, the person might still resist believing, because of how disturbing the thought is that they're wrong. If they have such a chip in their heads, that means that the apple they perceive in their hand was never an apple at all. That they were wrong the entire time. That there is actually a tortoise in their hand (and given how long the tortoise has been there through all this, it's probably gotten hungry and started chewing off a few fingers). It's easy to believe what you already believe, especially if your feelings back it up. Changing what we believe is tremendously uncomfortable, especially if our faculties lend themselves to what we already believe, and more so if the new way of looking at things makes us feel weaker, less intelligent, or simply duped. So for the most part, we try not to do it.
It's important to remember that depression (because we're talking about depression again) says nothing about how strong we are, how smart we are, or how good a person we are. All depression says is that we have depression. And what depression has done to us is make us see things in a way that's different than how they actually are.