Can't grab bezier control points

Is your mouse tracking set ridiculously high, so that moving your mouse a quarter inch causes your pointer to move across the entire screen?

My anchor selection tolerance is set at the minimum, but I have no problem hitting the target quickly.

If my suggestion about the preferences doesn't help then I just want to point out that there are adjustments in the "Selection & Anchor Display" section of the Illustrator preferences that might also help out. See screen shot:

I know what you mean. You have to really carefully watch the cursor, the change when hitting the handle is barely noticeable. But apart from that:

You can use this to make a wish http://illustrator.uservoice.com/

And another one: are you working with smart guides turned on? This helps a little bit.

Have you tried resetting your preferences? If something has worked in the past but is not working now this can often help restore what is lost.

To do this:

For Macintosh Users: With Illustrator closed launch a Finder Window in column view and click on your home folder. With the Option Key pressed choose Library from the Finder Go Menu. Within the Library folder find the following two files and delete them: “Adobe Illustrator <Version #> Settings” (earlier versions of Illustrator might just say “Adobe Illustrator”) and “com.adobe.Illustrator.plist”. When Illustrator is next launched it will create new preference files and the program will be restored to its defaults.

After you’ve reset up the program (make sure that no document window is open) to your liking, it is a really good idea to create copies of your personalized “mint” preference files (make sure that you quit the program before copying them—that finalizes your customization) and use them in the future to replace any corrupt versions you may need to delete.

For Windows Users: Hold down Ctrl + Alt + Shift when launching Illustrator and respond affirmatively when asked if you want to reset.