Wordperss Dashboard Not Loading Completely / API error?

I’ve never had this problem until recently. Only some of my stat widgets are loading in the dashboard. Sometimes everything loads; other times only a few. Here are some examples:

Stats (e.g., a chart of views per day): won’t load; or it will load but the chart will be missing.
Stats for Referrals, Search Terms, Post Views, Clicks: may or may not load.
Broken Link Checker (a plugin-dashboard widget): “[network error]”

(I’m also using WP Stats Dashboard, but it has always worked fine).

In a possibly, related error, I am getting an API error when I try to add or upgrade plugins.