Why can I only partially inflate my bike tyre?!?

You don’t say what kind or size of tyre. I have trouble getting my mountain bike tyres 2.3 x 27.5, above 20 psi with a hand pump.

Get an electric one that plugs into a car lighter socket.

You're doing it wrong, aren't you? You'll need to go to a bike shop and ask them to show you how to pump up a tire. Take your old tube, which is probably perfectly good, and your new pump and show it to them. 

It would have helped to know what kind of valve you’re dealing with.

Prestas sometimes needs to be ”burped” before they can be inflated.The actual moving part of the valve sometimes stick a bit, so hard that it can be difficult to get it to open with the pressure from the pump. That happens, you end up only pressurizing the hose.

The trick is to ”burp” the valve by hand before inflating the tube. Loosen the valve nut a few turns and depress the valve to let out a little air, check that it’s free to move. Then attach pump and proceed as usual.

This is a common problem. If the tube is nearly empty, use your thumb to push on the tire behind the valve and hold it tight into the rim valve hole. Then you should be able to get the valve deep enough into your hose connection clamp. 

If this is a Presta valve move the plunger stem nut close to the top but not all the way too the top

Two words...  "Operator Error".  You obviously... 1) You aren't seating the air chuck portion of the pump onto the valve far enough.  2) If it is on far enough, there should be a lever to move & lock it onto the valve stem.  3) You have a Presta valve stem & you're treating it like a Schrader valve stem or maybe vice-versa.  Pic below showing the difference between the two & a VERY good instructional YouTube video.  

https://www.youtube.com/watch?v=8AM0Zf49aVU    

It sounds like your pump isn't attached right. You need to make sure it's fully on the valve before you clamp it on or the air will leak out before going into the tire or the pump will not pump at all. Try a different pump