Aircraft rely on lift generated by interacting with the atmosphere and on using atmospheric oxygen to burn with fuel they carry.
Orbits aren't stable until you are high enough that there isn't enough atmosphere to interact with, and long before that the oxygen content drops too low to be useful.
So, to get to a stable orbit, you will need rockets eventually. Areospace types keep looking at hybrids (air breathing, lift generating while down low then switch to rockets as you get really high) because it does seem like a good idea, but so far the added complexity hasn't paid. Either the switching gear weighs too much, or the extra equipment adds too much risk of failure.
Maybe in the future.
Launching a space vehicle from a plane is an example of air launch and it does indeed provide some advantages like initial altitude boost and making greater part of the launch system reusable. Air launch has recently been gaining popularity, see Space Ship One's White Knight carrier plane and recent Paul Allen's plan for a new commercial spaceship.
However, going from this to a single vehicle with dual propulsion system (one for atmospheric flight and one for spaceflight) has very significant disadvantages. One of the greatest challenges when it comes to launching into space is the huge amount of fuel one needs to use for every kilogram of useful mass. The reason for this is that launching into space is not only about leaving the atmosphere (this isn't far at all: just 100km above your head), but it's also about achieving orbital or escape velocity (roughly 7-8km/s and 11.2 km/s respectively in the most common case). Accelerating a vehicle to this velocity requires huge amount of energy and fuel. See Tsiolkovsky rocket equation and this chart.
This challenge is the reason why launchers employ the opposite approach: unlike planes they jettison every heavy piece of equipment that is no longer useful for the remaining flight (e.g. first stages of a multistage rocket).
Carrying the atmospheric propulsion system with you all the way to space (and all the way to orbital or escape velocity) would make each launch extremely expensive if not impossible.
Depends on what you mean.
Get to high altitude? Yes you can. It is really inefficient to get there, but some aircraft already do it.
Achieve a stable orbit and be able to maintain control? No, at those altitudes you do not have enough oxygen to keep your engines running, you have to bring your own (and thus use a rocket)
For the sake of clarity, some definitions:
reaction engine: includes both rockets and jets. The engine pushes some mass in one direction and by reaction it achieves thrust in the opposite one.
jet (engine): is used to refer to airbreathing engines. The aircraft carries the fuel, but not the oxidizer; that is taken from the surrounding atmosphere.
rocket (engine): is used to refer to non-airbreathing engines. The (space)craft carries both the fuel and the oxidizer. No atmosphere is needed.
space: above 100km of altitude.
go to space:
staying in space:
Images from xkcd