It is unlikely there is more than one computer in the Batcave.
What we see are more likely to be terminals accessing a primary computer at a hidden location, either in the Batcave or someplace else more secure, somewhere else.
This would be a logical security feature protecting Batman's data and resources, requiring an access key available only to him and those who know it.
In the canon comic DC Universe
The Batcomputer is a supercomputer, whose specs are on par with any of those used by leading national security agencies, that permits global surveillance and also connects to a massive information network as well as storing vast amounts of information, both on his foes and his allies.
Depending on the DC continuity and Batman's predispositions, he may tap into resources of all the world's major intelligence agencies and is able to use official records from the FBI, NSA, CIA, Interpol, MOSSAD and MI-6 public and private resources.
He was also able to utilize the Department of Extranormal Operations records and at one point was able to utilize Brother Eye's data systems as well.
A series of satellite link-ups allows easy access to Batman's information network anywhere in the globe. The smart-systems are protected against unauthorized access, and any attempts to breach this security immediately sends an alert to Batman.
Depending on the DC Continuity and whether Batman is a team player, he may also connect his systems to Justice League computers which may use alien technologies and have resources the Batcomputer may not. He has had his technology connected to the Watchtower which used Thanagarian, Martian and Kryptonian technologies.
In the Dark Knight Series:
If we assume the Batcomputer is indeed a very powerful supercomputer, it is likely all you are seeing is a terminal aspect of the primary computer.
Given Batman's specialized needs, he would not be likely to have a supercomputer at every site, even if he could afford them, since they need care and maintenance of a specific nature to function properly.
It makes more sense for there to be one primary system and one secondary backup system at a remote location, should the primary be discovered or destroyed. If I Batman's budget, I would certainly purchase a building somewhere and fill it with the capacity to back up the primary Batcomputer in at least one other place.
The existence of these supercomputers are not to be confused with any number of instances (places where the Batcomputer can be accessed remotely) when we Batman sitting down to work on his the computer.
These instances can include his Batcave, any remote sites like his bunker, the computer access in the Tumbler/Batmobile, or even in a personal computer he may have in his costume accessing the supercomputer remotely by voice.
Production Notes
The user interface designer for the movie, The Dark Knight Rises, Jorge Almeida, discusses his design aspects given what he was told he would be working with in Flicks and Bits.
"“For ‘The Dark Knight Rises,’ I was told to keep the UI style real-world, which is obviously consistent with the style of the first two films. I read the script and was shown concept art for various sets, including the batcave and the fusion reactor. Both of these sets had a very sleek high-end style which was well constructed and designed, not hacked together."
Touching on his approach to the interfaces shown in ‘The Dark Knight Rises,’ and in particular the Batcave, Jorge Almeida had the following to say:
“The production designer, Kevin Kavanaugh, showed me production art for the batcave, which included a 9 monitor workstation which rose out of the ground. We discussed the scenes where the workstation would be used. He wanted it to look powerful without being slick or sci-fi. The challenge was to make it look interesting without feeling designed”…..”
When Bruce Wayne was doing research on Selina Kyle, the goal was to keep the UI from looking like a web browser. Gotham is an old city, so I thought using images of newspaper pages and arrest reports would suggest more of a historical archive than a website. I like mixing photoreal elements with the graphics when possible. I think it adds warmth to the design, and helps the audience register the information quicker.”
The Batcomputer was designed to appear as a remote visual interface. You could think of it as a browser interface for his supercomputer. A Chrome laptop might be an apt description. There is no operating system only a connection to a data source. The Batcomputer is the source and where ever Batman is accessing the system becomes the terminal he uses.
This meant any time you saw the Batcomputer in action, you were likely seeing an instance of the main computer running from a remote terminal. This makes sense. It means Batman only has one supercomputer, likely in the Batcave with a redundancy in another location as a backup. Any sites he uses remotely like his bunker would connect to his main computer without having to store any files or information on those machines as a security feature.
Batman logs in. He works, he gets whatever information from the Batcomputer (where ever the actual supercomputer resides) and then he logs off. No chance of any data being found or recovered because nothing is stored on site.
This setup makes sense considering Batman's security needs, he can't afford to leave data trails lying around. There is never any data stored anywhere local.
Without hard drives to worry about and as long as he destroys the equipment properly no way to trace the data back to its source system (which could be further obscured using the telecom systems of the planet like hackers do to hide their activities).
In that way he could have as many terminals (places where he can access the Batcomputer) as he desires in any facility connected to his private high speed network.
