With less expensive guitars, string height off the pickup is another thing to consider, especially if you use light guage strings.
If you know you are staring off low when trying to tune the high E, and you just keep going until it breaks but the tuner never seems to pickup the tone from the string... sometimes the pickup is too low. Cheaper pickups in less expensive guitars tend to have less response than more expensive ones and the sometimes it helps to bring it up a little and get it closer to the strings.
If your pickup is set in too low (top of the pickup flush with the body of the guitar), grab a screw driver, hold down the E string at the highest fret and jack up the pickup until the string is about 1/16th inch away from the magnetic pole, or screw of the pickup.
This may or may not help. And, yes, you will see many guitar players with their pickups set in and flush with the body of their guitar. Usually for those situations, the neck and body are not very far apart in height (pic above), or these tend to be "hotter" pickups, meaning they have a higher output and typically use stronger magnets to pickup the vibrations from the strings.