If you hear an interval wrong, you will most likely play it wrong. Many people play octaves too wide and it's not because their instrument is out of tune; the problem is mostly in the brain.
A good trick to fix that is to play a note with a tuner and sing it in your head at the same time and then close your eyes and play an octave above or below while still singing the first note in your head and trying to tune to the pitch you are producing in your brain. Once you think you're in tune with your brain, open your eyes and check if the tuner agrees with you. You should be much closer at least!
Do this exercise on many notes for a few days and you should see some improvement.
You can change the intonation of a note by changing the tone color. If you make a brighter sound, your pitch will tend to go sharper and a darker tone should help you going lower. This can be very helpful in the low register where the pitch can vary a lot. It is more difficult to do this in the high register but it still makes a difference.
Some people say we should never use an electronic tuner. I think tuners can be very useful if used correctly.
It's not a good idea to clip a tuner on your instrument and tune every note on sight while playing in a group. It's more important to tune to the group than to a machine so you'll have to learn to do it by ear . With a bit of practice, it's not that hard. Also be aware that notes are tuned differently depending on their function in the chord. ( major thirds are tuned lower, minor thirds higher etc.) So when you are working on brass chords, you can only use the tuner for the root of the chord. The tuner can be used if we disagree on some notes but not all the time.
When teaching a group of beginners or young students, it's easy to lose the pitch when the kids are going in all directions. A quick check on the machine will help putting the group back on track.
It's also very useful when you need to arrange your or a student's slides.
Intonation can be a very subjective thing. Depending on how our ears have been educated, we ear things differently. The way we "feel" the pitch is also different from one person to another. I played an exam once where one of the three judges said I was a bit flat, the other one said I was a bit sharp and the last one said I was just right! That happens a lot actually.
I personally prefer someone using an electronic tuner once in a while to someone who pretend he hears everything but doesn't really have a clue. There's nothing wrong in admitting you don't hear everything like a machine. As long as you can tune yourself by ear in the group, it's ok to take a couple of seconds to check a note here and there with the machine.
Don't be too precious to about your ear. At the end, it's not about who's right or who's wrong. It's about how do we play in tune together. If tuners can help us a little, than why not?
When you sing the music in your head while playing, your brain gives orders to your lips, fingers and the rest of your body to produce what is playing in your head. It will send signals to your muscles and after a few times playing like this, will find the most efficient way to do it. Of course you have to know your music. We very often say: "if you can sing it, you can play it" . It is true assuming that you have the strength to reach the notes of course.
I personally find after a while doing this that my lips start feeling very good and my tone improves a lot. I also have more endurance. I believe the reason for that is that the brain is acting to produce the desired notes and sounds rather than reacting to what it hears. The brains no longer needs to correct each note's tone and pitch after it hears them thus saving energy to play longer.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.