Some of the blocks are very quick to create. These are included in the page.
Some of the blocks are slow to create. These are loaded after the main page has loaded.
Otherwise, the main page would take a long time to load.
Your home page has three "slow" blocks.
On this day - 20 seconds.
Top surnames - 18 seconds.
Statistics - 33 seconds.
I guess that when your server is busy, these blocks take longer than your server's maximum CPU time (typically 60 seconds).
This would explain why blocks do not appear.
Here are two things you can do to make things faster:
* For the statistics block, you can remove the "top surnames". This information is duplicated in the block below.
* For the statistics and top-surnames blocks, the data does not change very often. You could copy/paste this into an HTML block - which would load instantly. You would need to update it manually every week or month.