Losing sleep after someone dies is definitely a thing. But it wasn't my thing.
Yes, sleep in the early days were near impossible. I had to put on a TV show in the background..a show that was not too funny, but funny enough. A show that was comforting and familiar, but had no associations with my sister (I chose How I Met your Mother). The volume had to be low enough to go to bed, but loud enough that I could just make out the dialogue. The goal was to give my destroyed brain a moment of respite so that it could slip into unconsciousness.
This didn't stop me from waking, usually several times a night. Both because I would be startled from sleep and instead of adjusting my pillow, I'd remember my sister was dead and I would sob, or because my daughter was still nursing and she would cry for me, and I walk to her and remember my sister was dead and I'd sob.
Within a couple of weeks, my sleep schedule was close to normal. The wakings in the night no longer left me crying into my pillow for hours at a time, but I was still so tired. Bone deep tired. Barely functioning tired.
I think that the bereaved are not tired because of lack of sleep, it actually has much more to do with all that happens when we're awake. 
For more (trying to) Explain the Inexplicable - What Grief Feels Like and What it Feels Like When Your Sibling Dies Unexpectedly.
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