Blocks can't be broken properly -- they reappear immediately after breaking

I hadn't played Minecraft for a while. Firing up the game again, it downloaded the latest version (1.9.2). I created a world, and attempted to collect resources. I discovered that when I try to destroy a block, it immediately reappears, not dropping anything. If I wait a while (say around 5 seconds or so), suddenly the block disappears again leaving a drop. This of course makes the game all but unplayable.

Note that I recently upgraded my graphics card and added more memory. I mention this in case there's a cache some place that needs to be refreshed.