I disagree with that. In this day and age there’s almost no way to NOT write down passwords.
I see it this way: when you do write it down (assuming, of course, you store everything in a reasonably secure place…like your wallet) it makes it much easier to choose GOOD passwords, as opposed to lousy one’s just so you can remember them…

Btw., most authentication routines use 128-bit encryption/hashing, incl. Linux passwords (MD5). In order to make your passwords at least as strong as the underlying algorithm you need at least 28 *randomly chosen* characters, if you were only picking from the 26 lower-case english alphabet. Picked from upper and lower-case (52 chars) you’d need 23 randomly chosen characters and if you add the 0-9 numbers into the mix (62 chars), you’d still need at least 22 random characters out of that.
Now you know how weak your password really is compared to the technical implementation of it :-)