After receiving a message from a boy, telling me the bank is being robbed, I ran to the bank to speak with Velerad. I waited for Siegfried to arrived, and agreed to enter the bank's back door together. Siegfried ran off into the bank, but I decided to leave the quest for later.

Many quests later, I decide to finally go ahead with the Gold Rush quest, but unfortunately, I cannot enter the bank at all; all doors are locked. Why is this? Is it possible to progress without this quest?

This problem has now been solved

run around to the back ( near the medallion on the wall) see if Velerad is there and if you might be able to use a different , nearby entrance. Not saying more, good luck
PS it's a major plot quest, pretty sure you will have to find one door or another to get in.

Thanks for the reply. As I stated, all doors are lock. Front, back, and warehouse door - all locked. If I speak to Velerad, he says something along the lines of "Help, the bank is being robbed", then the scene ends.

Granted, the people who visit this forum have probably already played through at least once, but I am loath to post spoilers, just in case.

Since you've said you finished other quests first, I have to wonder which ones you completed in chapter three, so far.

Depending on which ones you have completed, certain doors will be locked. There are three ways into the bank. The door on the corner where the knight hands you the pass to hunt kikimores. The one on the other side next to where the sensor is placed, and the warehouse. There is a way through the sewers to get to the basement of the bank, but it won't let you go all the way to the first level.

There is a walkthrough and consequence discussion on the wiki @ http://witcher.wikia.com/wiki/Gold_Rush but if you don't want spoilers, don't go there.

I appreciate all your help. I was able to fix this issue through some unconventional means. If I were to guess I'd say because I chose not to enter the bank (Town Hall) immediately after Siegfried, the door returned to being locked. This meant I had to obtain the keys to either the warehouse door or the town hall.

Fortunately, with the help of a third party save editor, I was able to add both these keys to my inventory, allowing me to go forward with the mission.

Thanks again for your support.

Glad you found a solution that worked.