Whether "but" is used correctly?

I suspect that in this case the original writer has used the word "but" correctly, however the sentence then only makes sense if the second clause is modified to make it a positive attribute. I'd suggest the intended meaning was actually,

"Company databases are not immune to crashes or failures, but they DO have the luxury of downtime." (losing the not and DO is my emphasis).

This makes sense semantically from a knowledge of how (some) company databases work. Your proposed modification also makes formal sense and would be a reasonable correction if we assume company databases should never have downtime. This is actually an arguable point depending on the kind of database so both modifications are possible but you are right to notice that the original sentence is structured incorrectly.

I believe that the use of "they" is ambiguous. While databases are not immune to crashes or failures, it is companies that do not have the luxury of having database downtime. The use of they implies that it is the databases themselves that do not have the luxury of downtime, which doesn't sound quite right to me. Here is how I might compose that sentence to be as clear as possible:

Companies do not have the luxury of downtime, and their databases are not immune to crashes or failures.

The conjunction "but" is used to join contrasting thoughts. The two clauses here express contrasting thoughts. Therefore, it makes sense to use "but."

The first thought is that databases may crash or fail. An implication of this fact is that databases will not be available all the time. That implication contrasts with the thought expressed in the following clause.

An analogous sentence: my raspberries are not growing slowly in this sunny weather, but I do not have the luxury of sufficient time to pick and preserve them.

“But” signifies a contrast. So whether it is appropriate depends on whether the two clauses have a point of conflict. For example, consider the following pair:

• Water is precious.
• Everybody needs water.

We can set up a point of contrast by arguing for conflicting answers to “Should we drink water?”, depending on which proposition we pick. So we can connect the two with “but”:

• Water is precious, but everybody needs water.

In your example, here are the propositions:

• Company databases are not immune to crashes or failures.
• They do not have the luxury of downtime.

We can set up a point of conflict by asking, “Do we need to budget for downtime?” The first statement requires it; the second doesn’t allow for it.

As such, linking the two clauses with “but” works.