We know that Gryffindor Tower uses a password system: one provides the password and is granted access. A non-Gryffindor using the wrong password may be against school rules, but is perfectly possible. Indeed, we frequently see Neville locked out because he has forgotten the all-important password. Yet, when Sir Cadogan lets Sirius Black in with the password, he is fired. Why?

Theory #1: he should have recognized Sirius Black. But why? If the portraits can recognize people, why are passwords used at all? Subsequent to this incident, Flitwick apparently taught the front doors to recognize Black, but no one mentions training the common room portraits.

Theory #2: he should have been suspicious that Sirius read the passwords off a sheet. And yet, the Fat Lady clearly witnessed Gryffindors telling each other the password and getting in. Further, Neville has been using a list of passwords for some time.

It may have been his fault.

Sirius Black was able to gain entry to the common room by using a stolen list of passwords. But why would someone have such a list? We are actually told why, earlier in the story back in Chapter Twelve:

"I've lost the passwords!" Neville told them miserably. "I made him tell me what passwords he was going to use this week, because he keeps changing them, and now I don't know what I've done with them!"

We see that Neville only had to write down the passwords because Sir Cadogan continually changed them. Under a regular portrait's reign, a single password could be operative for weeks at a time. For example, in Goblet of Fire the password at the beginning of the term was "balderdash", and that was still the password several months later in December. From Chapter Twelve:

"Password?" she said as they approached.

"Balderdash," said George, "a prefect downstairs told me."

Then in Chapter Twenty-One:

The start of December brought wind and sleet to Hogwarts.

"Wonder where she's got to?" Ron said as he and Harry went back to Gryffindor Tower.

"Dunno . . . balderdash."

But the Fat Lady had barely begun to swing forward when the sound of racing feet behind them announced Hermione's arrival.

Thus, it could be argued that Sir Cadogan was principally to blame for the break-in. Had he not had such deviant password policies Neville would have never had to write the passwords down, so Sirius would never have been able to gain entry to the common room with a stolen list.

Sir Cadogan likely wasn’t fired for Sirius.

Though Professor McGonagall did question him about if he let Sirius in, she only seemed upset with Sir Cadogan when she thought he let Sirius in without the password. After she learned that Sirius did have the passwords, she was instead angry at the student who wrote them down, Neville Longbottom.

“Sir Cadogan, did you just let a man enter Gryffindor Tower?’

‘Certainly, good lady!’ cried Sir Cadogan.

There was a stunned silence, both inside and outside the common room.

‘You – you did?’ said Professor McGonagall. ‘But – but the password!’

‘He had ’em!’ said Sir Cadogan proudly. ‘Had the whole week’s, my lady! Read ’em off a little piece of paper!’

Professor McGonagall pulled herself back through the portrait hole to face the stunned crowd. She was white as chalk.

‘Which person,’ she said, her voice shaking, ‘which abysmally foolish person wrote down this week’s passwords and left them lying around?’

There was utter silence, broken by the smallest of terrified squeaks. Neville Longbottom, trembling from head to fluffy-slippered toes, raised his hand slowly into the air.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 13 (Gryffindor versus Ravenclaw)}}

After that, it’s clear that Neville is the one Professor McGonagall faults for Sirius being able to break in. Neville is punished severely, and isn’t allowed to know the passwords.

“Neville was in total disgrace. Professor McGonagall was so furious with him she had banned him from all future Hogsmeade visits, given him a detention and forbidden anyone to give him the password into the Tower. Poor Neville was forced to wait outside the common room every night for somebody to let him in, while the security trolls leered unpleasantly at him.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 14 (Snape’s Grudge)}}

It doesn’t seem like McGonagall considered Sirius being able to break in Sir Cadogan’s fault - he let Sirius in because he had the passwords. Instead, she placed the blame squarely on Neville for keeping a list of the passwords.

His post was probably always temporary.

Though Professor McGonagall was surprised to learn that Sir Cadogan let Sirius Black into Gryffindor Tower, the reason he was ‘fired’ may simply be because the Fat Lady was finished being restored, and therefore able to take her job back. Sir Cadogan being removed as the guard for Gryffindor Tower coincided with the Fat Lady being fully restored and returning to her job.

“Sir Cadogan had been sacked. His portrait had been taken back to its lonely landing on the seventh floor, and the Fat Lady was back. She had been expertly restored, but was still extremely nervous, and had only agreed to return to her job on condition that she was given extra protection.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 14 (Snape’s Grudge)}}

Sir Cadogan was never the optimal choice for Gryffindor Tower’s guard. When the Fat Lady was attacked, the only reason he got the job to begin with was because he was the only one who was willing to take it.

“The Fat Lady’s ripped canvas had been taken off the wall and replaced with the portrait of Sir Cadogan and his fat grey pony. Nobody was very happy about this. Sir Cadogan spent half his time challenging people to duels, and the rest thinking up ridiculously complicated passwords, which he changed at least twice a day.

‘He’s barking mad,’ said Seamus Finnigan angrily to Percy. ‘Can’t we get anyone else?’

‘None of the other pictures wanted the job,’ said Percy. ‘Frightened of what happened to the Fat Lady. Sir Cadogan was the only one brave enough to volunteer.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 9 (Grim Defeat)}}

Therefore, it seems highly likely that he was always going to be taken off the post once the Fat Lady was able and willing to return to her place guarding Gryffindor Tower.

He shouldn't have given Neville the passwords in advance.

In fact, he probably shouldn't have given Neville the passwords at all; from what little we see of the Gryffindor common room's so-called security system it appears to be based around the Gryffindor prefects - the painting tells them the password, and they tell the other Gryffindors.

"Password?" she said as they approached.

"Balderdash," said George, "a prefect downstairs told me."