Do you ignore friends that are having a hard time with a mental health condition?
I prefer the honest approach of 'I'm sorry but I can't deal with this right now. I'm a bit pre-occupied with my own issues.' There's nothing wrong in saying this. It's honest but it doesn't say 'I'm cutting my ties with you'.
The other side of the coin is that there are people that will break communications with you through fear based on stigma. If that's the case, are they the sort of friend that you needed in the first place?
Friendships require trust and effort, the same as any kind of relationship. They don't grow on trees. No one rubbed a lamp to make one appear.
Every now and then I recognise that a friend or acquaintance is in need of some help and support. I can't give much as I have to balance my own energy and cope with my issues. I'd never turn my back on them though.
If you know friends that have such issues and are at a low ebb, please, don't ever abandon or ignore them. Be realistic with your own resources ... but remember that you called them a friend for a reason and they might need your help right now.