Upgrading to  Drupal 7.7 produces a fatal error: What can I do?
The database.inc file does contain the definition of the class DatabaseConnection.

/**
 * Base Database API class.
 *
 * This class provides a Drupal-specific extension of the PDO database
 * abstraction class in PHP. Every database driver implementation must provide a
 * concrete implementation of it to support special handling required by that
 * database.
 *
 * @see http://php.net/manual/en/book.pdo.php
 */
abstract class DatabaseConnection extends PDO {
  // …
}


It is probable the file was not correctly copied, or has been corrupted during the copy; that would explain why the class has not been found, but the class is effectively present in the file that comes with Drupal 7.