got down on

[Ashiuhto]

Please check the following paragraph.

Lin's family always believed in him and picked him up when he could have gotten down on himself. That made him continue to believe. If you want your family to believe in you like that, you've got to be there for them too when they need it.

[Raymott]

Please check the following paragraph.

Lin's family always believed in him and picked him up when he could have gotten down on himself. That made him continue to believe. If you want your family to believe in you like that, you've got to be there for them too when they need it.

This phrase doesn't sound right to me. I assume you mean he could have become depressed or given up looking after himself, or trying to cope.
I'd suggest "...when he could have given up on himself", but someone else might recognise the phrase you've used.

[SoothingDave]

I recognize the phrase.

[Tullia]

As do I, but I would say it was very informal, and perhaps even sounds a little uneducated. I think that's because of "gotten" which I always find a very ugly word! To my ear it would sound a little better with "been" instead.

I also think it's more AmE than BrE. (As Lin is American, that's probably not an issue though.)