What process is stupidly complicated or slow because of "that's the way it's always been done" syndrome?

If they’re still doing bayonet training-when was the last bayonet charge, WW2? They were outdated in WW1!

Contracting for anything. Whether it's big or small dollar-wise, the contracting process in very complicated. I've seen it both from the contractor point of view and the contracting officers' representative viewpoint. Contracts for services are difficult to get funded and pushed through the contracting system in both the Air Force and Army. I suspect the Navy is the same.

One of the biggest problems is the number of approvals you need to have at each phase of the contracting process. I won't recount all of the issues I experienced here, but will highlight a few big problems:
-Funding: justification for a services contract can take a year or more. If you do get the funding in your unit's budget, it can still be raided for other priorities. About the only way to prevent this is Congress appropriating money for a specific project. They usually only do this for major acquisition programs like a new aircraft or tank or ship. If you just need a $1-million services contract to support a program, you may only get $800K by the time the money filters down to your level.
-Writing the Statement of Work: Two problems here are Contracting Officers (KOs) who don't understand your business and Contracting Officers' Technical Representatives (COTRs) who don't understand acquisition. The mutual ignorance often places these folks at loggerheads when it comes to writing the statement of work (SOW). A good COTR will educate the KO on their job enough for them to be helpful. A good KO will allow the COTR the maximum flexibility in explaining the work they need the contractor to do within the limits of the contracting regulations. Too often each party doesn't want to take the time to do it right. When this happens, the prospective contractors are stuck trying to bid on a statement of work that is incomplete, poorly written, and doesn't provide the Services the COTR really needs. For example, the KO may require the SOW to contain a strategic-level objectives statement that generally states the reason for the contract and the work to be done. The COTR may provide that statement and then neglect to provide details of the work that supports each element of the high-level statement. The contractors will then have to submit many questions to the KO during the bid and proposal process to attempt to get the statement of work modified into something the contractors can actually bid on.
-Low-cost (minimum) qualified selection: I ran into this one as an Army Civilian. We were forced to accept the lowest cost bidder who was minimally qualified to do the work. The "best value" bidder was usually not the low-cost bidder. Consequently, we awarded contracts to companies who were minimally qualified, i.e., they weren't unqualified, to do the work. Then the COTR and other functional people ended up training the contractor's employees on the work they should have known how to do the day they showed up.

There's much more to be said about contracting, but I'll leave it there because this post is too long already.

But they keep incompetents from making things worse