Some computers connect, some don't. Router? Cable modem?

Apple desktop fine. Laptop iffy. Kindle "sees" network and says "connected", but won't work (I get a little "x" next to the full-strength wireless signal icon).

Is Time Capsule being squirrley? The Kindle people tell me to reset the DNS on my router? What on earth does that mean? I've already power-cycled several times.