Can you establish the canon using the bible alone?

The bible does in fact give us enough information to establish the canon. 

Firstly, 2 Peter 3:16 calls Paul's writings canonical: "As also in all his epistles, speaking in them of these things; in which are some things hard to be understood, which they that are unlearned and unstable wrest, as they do also the other scriptures, unto their own destruction.".

From this we can see that the writings of Paul are canonical, but how do we also establish 2 Peter?
Jude seems to actually quote 2 Peter as an authority: "17 But, beloved, remember ye the words which were spoken before of the apostles of our Lord Jesus Christ; 18 How that they told you there should be mockers in the last time, who should walk after their own ungodly lusts." This can be compared to 2 Peter 3:3 saying "Knowing this first, that there shall come in the last days scoffers, walking after their own lusts,". Critical scholars believe that 2 Peter is quoting Jude, however the wording of Jude implies that he was quoting the apostles, Paul made similar predictions but the wording of Jude is clearly taken from 2 Peter.

We can also argue that Jude is canonical as he was a brother of Jesus and spoke authoritatively.

 Thus our canon now includes:
Jude, Peter's letters and Paul's letters. 

Next Paul calls the Jewish bible as inspired in Romans 3:2 saying that the Jews were given the canon: "Much every way: chiefly, because that unto them were committed the oracles of God."

Paul also called the book of Luke canonical in 1 Timothy 5:18: "For the scripture saith, thou shalt not muzzle the ox that treadeth out the corn. And, The labourer is worthy of his reward." taken from Luke 10:7 "And in the same house remain, eating and drinking such things as they give: for the labourer is worthy of his hire. Go not from house to house."

Paul calling Luke inspired also implies Acts being inspired, as we know that they have the same author from identical style and this being spelled out in Acts 1:1 directly "The former treatise have I made, O Theophilus, of all that Jesus began both to do and teach,"

Now our canon includes Jude, Peter's letters, Paul's letters, the Old Testament, Luke and Acts.

Next we can imply that Matthew and Mark should be canonical as it is clear Luke made use of both.

Mark and Matthew also had apostolic authorship.

For the book of Revelation it has debated authorship, Justin Martyr believed it to be John the apostle while Eusebius believed it to be a long time companion of the apostle called John the Elder. However in both cases the writer was an apostolic witness, and when he speaks for Jesus it can be trusted. It can also be established canonical from the fulfilled prophecies it has, as it prophesied the persecution of Smyrna that happened in 155ad. The book states: "Fear none of those things which thou shalt suffer: behold, the devil shall cast some of you into prison, that ye may be tried; and ye shall have tribulation ten days: be thou faithful unto death, and I will give thee a crown of life." Though there were immediate persecutions, the one that came in 155ad was strong and directly documented.