Why can I sometimes not change weapons as a Heavy

I know that the Heavy requires some cool-down time (before switching weapons) after spinning his Minigun, but, sometimes, even if I wait a good 10 seconds, I still get the can't-change-weapons sound, which is very frustrating when I'm in a situation where I either need to eat my Sandvich or make a quick escape with the GRU. Why is this, and how can I ensure that I can switch from my Minigun to my Sandvich as quickly as possible?

I have also noticed that, if I try to change weapons immediately after spinning my Minigun, it takes significantly longer for me to finally switch weapons.

Answers to Meme Scientist's questions, to provide more context:

• Are you playing in medieval mode, or a gamemmode which prevents the use of guns?

  No, I am playing on normal mode.

• Are you playing on Valve servers or community servers?

  About 50-50. This happens far more often on community servers like Saigns.de, though (although I'm not sure what the server has to do with my inability to switch weapons). I should also note that my latency on all of these servers is around 30 ms (any higher, and I don't bother playing on that server).

• What are your keys bound to? It's possible you may have rebounded your quickswitch or individual weapon selection switch to another key.

  • MOUSEWHEELUP := weapon 1
  • MOUSEWHEELDOWN := weapon 2
  • MOUSEBUTTON3 := weapon 3 (melee weapon).

• What weapons cause this issue? Do all miniguns cause this issue (ie Huo Long, Tomislav, Brass, etc)

  This happens with all miniguns (and is independent of my secondary and melee weapons).

• Does the same issue occur in online and offline gamemmodes? Latency could be a huge factor; maybe you are lagging on a server, causing you to see that it takes nearly 10 seconds to switch weapons

  No, they do not, but, as I've mentioned, I only play on servers where my ping is below 50.

Given this information, what could possibly be causing my inability to change weapons from a Minigun?