How do I tell if a regular file does not exist in Bash?

Negate the expression inside test (for which [ is an alias) using !:

#!/bin/bash
FILE=$1

if [ ! -f "$FILE" ]
then
    echo "File $FILE does not exist"
fi

The relevant man page is man test or, equivalently, man [ -- or help test or help [ for the built-in bash command.

Alternatively (less commonly used) you can negate the result of test using:

if ! [ -f "$FILE" ]
then
    echo "File $FILE does not exist"
fi

That syntax is described in "man 1 bash" in sections "Pipelines" and "Compound Commands".

The test command (written as [ here) has a "not" logical operator, ! (exclamation mark):

if [ ! -f /tmp/foo.txt ]; then
    echo "File not found!"
fi

Bash File Testing

-b filename - Block special file
-c filename - Special character file
-d directoryname - Check for directory Existence
-e filename - Check for file existence, regardless of type (node, directory, socket, etc.)
-f filename - Check for regular file existence not a directory
-G filename - Check if file exists and is owned by effective group ID
-G filename set-group-id - True if file exists and is set-group-id
-k filename - Sticky bit
-L filename - Symbolic link
-O filename - True if file exists and is owned by the effective user id
-r filename - Check if file is a readable
-S filename - Check if file is socket
-s filename - Check if file is nonzero size
-u filename - Check if file set-user-id bit is set
-w filename - Check if file is writable
-x filename - Check if file is executable

How to use:

#!/bin/bash
file=./file
if [ -e "$file" ]; then
    echo "File exists"
else 
    echo "File does not exist"
fi 

A test expression can be negated by using the ! operator

#!/bin/bash
file=./file
if [ ! -e "$file" ]; then
    echo "File does not exist"
else 
    echo "File exists"
fi