Asking a Yes/No Question from a Bash Script

I have to warn you, I'm really not very much fun at parties, simply because I take my shell scripting seriously.
There's a bunch of bad stuff in your code so please don't take my criticism the wrong way. I am trying to be constructive.

#!/bin/bash
# First thing to note is that this is a bash script. Your script seemed to be running something called bash.sh
# I have no idea what that was.

# Set up warning and error. Why NOT make error call warning. Also note that there is never a good reason to refer
# to $*. Instead, always use "$@". And the cost savings of using >&2 instead of 1>&2 is just not worth it.
warning() { echo "$@" 1>&2; }
# Also, let's not forget that simple commands can be stacked into one command line.
error() { warning "$@"; exit 1; }

# Here we set usage up so that we can die with a bit more elegance.
usage()
{
    warning "$@"
    cat <&2
Usage: $prog [--timeout val] [--default yes|no] prompt
EOF
    exit 1
}

#
# Usage: yesno OPTIONS QUESTION
#
#   Options:
#     --timeout N    Timeout if no input seen in N seconds.
#     --default ANS  Use ANS as the default answer on timeout or
#                    if an empty answer is provided.
#
# Exit status is the answer.

# One notion is to declare things to be of type int.
# Also, never do your own options parsing. It's really hard and why invent the wheel? 
yesno()
{
    local ans
    local -i ok=0
    local -i timeout=0
    local tv	# TimeoutVal
    local default
    local temp
    # BTW, NEVER give a variabel a one-letter name. It's hard to find them using any editor.

    temp=$(getopt -o :d:t:h --long default:,timeout:,help -n $prog -- "$@")
    eval set -- "$temp"        # The double quotes are key here.
    shopt -s nocasematch       # Hey, this is bash. Let's not make life harder for ourselves.
                               # None of this checking to see if it's equal to y or to Y.
    while true
    do
        case "$1" in
        -d | --default)
            # No need to shift 1 to then get the value and then shift again. Just ref "$2" and then shift 2.
            default="$2"
            [[ -z "$default" ]] && error 'Missing default value'
	    [[ "$default" != y  
           &&  "$default" != yes
	   &&  "$default" != n
	   &&  "$default" != no ]] && error "Illegal default answer: $default"
            shift 2
            ;;

        -t | --timeout)
            # Note that tv is a string but if we decide that it's safe to assign to an in var then we do it.
	    tv="$2"
            [[ -z "$tv" ]] && error 'Missing timeout value'
	    [[ "$tv" =~ ^[0-9][0-9]*$ ]] && timeout=$((tv)) || error "Illegal timeout value: $tv"
            shift 2
            ;;

	-h | --help)
	    usage "Here's what you need to do to run this"
	    ;;

        --)
            This is how e get out of the options processing game.
	    shift
            break
            ;;

	*)
	    usage "$1: Unknown option"
	    ;;
	    
        esac
    done
    # Now that we're done with the procopt then we can do the consistency checks.
    if (( timeout != 0 ))  &&  [[ -z "$default" ]]
    then
        error 'Non-zero timeout requires a default answer'
    fi

    (( $# == 0 )) && error 'Missing question'

    while (( ok == 0 ))
    do
        if (( timeout != 0 ))
	then
            if read -t $timeout -p "$@" ans
	    then
                # Turn off timeout if answer entered.
                timeout=0
                # No need for all this testing and then assigning stuff. Just do it.
                : ${ans:="$default"}
            else
                ans=$default
            fi
        else
            read -p "$@" ans
            : ${ans:=$default}
        fi

	[[ "$ans" = y  
	|| "$ans" = yes
        || "$ans" = n
        || "$ans" = no ]] && ok=1 || warning 'Valid answers are: yes y no n'
    done
    [[ "$ans" = y || "$ans" = yes ]]
}

prog=$0
# This is just so you can see that you don't need basename and dirname. This is bash, not sh.
fn=${0##*/}
noshext=${fn%.sh}
# Also, not that I'm testing yesno, not 'yesno' and not "yesno". Don't interpolate if you don't have to..
if [[ ${noshext} == yesno ]]
then
    yesno 'Test bad timeout value? ' && yesno --timeout none "Hello? "
    yesno 'Test timeout without default value? ' && yesno --timeout 10 'Hello? '
    yesno 'Test bad default value? ' && yesno --default none "Hello? "
    yesno 'Yes or no? ' && echo 'You answered yes' || echo 'You answered no'
    yesno --default yes 'Yes or no (default yes) ? ' && echo 'You answered yes' || echo "You answered no"
    yesno --default no 'Yes or no (default no) ? ' && echo 'You answered yes' || echo 'You answered no'
    yesno --timeout 5 --default no 'Yes or no (timeout 5, default no) ? ' && echo 'You answered yes' || echo 'You answered no'
    yesno --timeout 5 --default yes 'Yes or no (timeout 5, default yes) ? ' && echo 'You answered yes' || echo 'You answered no'
fi

So, in the end, the goal is to write bash code and not Bourne shell code. Make it simple to read.

What common mistake does this script help to avoid? Your 'this common mistake' link is just a picture to a gun pointing at a foot.

That code is horrendously complicated for something that should be simple. For example, why test for 'yes' and 'no' when you could force a one-character user input by passing '-n 1' to the read command? Secondly, use getopt or getopts to parse arguments - gives more flexibility. Also why allow the user to make mistakes by allowing them to specifiy arbitrary defaults? Specifying the default by passing '-y' or '-n' means it's easier to use and easier to write (no need to check for invalid defaults - handled automatically by getopt).

Nice, but bash is so confusing. I'll stick to python.

The following:

if [[ "$t" != 'y' && "$t" != 'yes' && "$t" != 'n' && "$t" != 'no' ]]; then
error "Illegal default answer: $default"
fi
Can be more clearly put as:

case "${t}" in
y)
yes)
n)
no) ;;# acceptable
*)
error "Illegal default answer: $default
esac

Your use of $* over $@ shows that you have still many bullets left.