I've read about instances of class changes in the middle of a campaign, but what about background changes?
For example, if a level 5 character had a Soldier background, but found another to fit their backstory more, would they be allowed to change it?
Are there any rules about this?
You can't by RAW
I believe that 3.5 might have had some rules about retraining, but 5e definitely does not have any rules about changing classes (maybe you're thinking about multiclassing?)
The PHB (pg. 126) states that
Your character’s background reveals where you came from, how you became an adventurer, and your place in the world.
To change this would mean changing the character's history. Maybe a wish (or some other reality altering effect) could cause this to happen?
You can probably do it anyway
That being said, everything's up to DM discretion anyway. I've let my players retcon all sorts of different stuff, including class, backgrounds, feats, etc., and my DMs have let me retcon my characters without too much hassle. If you're a player, talk to your DM. If you're a DM, I don't think too much will be lost by letting your players change their characters, especially if it's because they decided that another background fits their character's backstory better.
Backgrounds have a few purposes in 5e. From a character development standpoint, they define who they were prior to becoming Joe the Wizard or Bob the Fighter. In this way, they help to flesh out characters more fully, by helping to give them a place in the world they live in.
From a game mechanics standpoint, backgrounds give a character proficiencies appropriate to the background as well as some thematic, loosely defined ability that let's them excel in some non-combat manner.
To directly answer your question about changing a level 5 soldier's background, the short answer is not without DM's permission and essentially DM's fiat. This makes sense from both a story perspective and game mechanics perspective. For 5 levels now, that character has been presenting himself as a soldier, so it'd be odd if they suddenly were not. Also, for 5 levels, they have been using the skill proficiencies granted by having the soldier background.
If the character's goal is to shift their ideals, flaws, etc., there is some basis for that in both the PHB and DMG. These are underlying traits of a character that provide guidance on how the character interacts with the world, but they're not mandatory rules. On page 125 of the PHB, there is information to personalize a background if nothing is a perfect fit. It may be prudent to utilize this and consult with your DM if you want to change some of these aspects of your character. Also, there are periodically effects in the game which make your character gain a new flaw, but I suspect this may be beyond the scope of what you're asking.
Adventurers League allows complete rebuilding of characters until they reach level 5. The player keeps the character's name, experience and treasure, but can change anything and everything else.
I think it would be reasonable for a GM to allow this in a private game.
From the Players Guide:
The character keeps all experience, treasure, equipment, magic items, downtime, and faction renown earned to that point. The character replaces the old starting equipment (along with any gold earned from selling it) with the new starting equipment. If a character’s faction is changed, that character loses all renown earned with the former faction, and starts at 0 with the new faction.
Whatever you want.
I think the other answers have given you some good info, so I'll try to have my own take on it.
I like to try to make things work out with their current backgrounds. If they picked a canned background and decided they wanted more flair instead, or maybe they started with a shallow one and wanted more complexity; either way, I would allow them to do it. Whatever makes things more fun for the players (without ruining fun for others) should be your goal as DM.
I've had this happen a few times with players, especially new ones. As players get more comfortable with each other or more experienced doing this sort of thing, they want to fix the mistakes their previous selves made.
I will also add that when possible, I add little side quests to "retcon" things. I had one player who, playing a druid, decided it didn't make sense that he all of a sudden just knew how to beast shape. I made a little side quest for him so that while under duress, he just shifted into a creature. Since then he has had control over it. Slightly different than changing a background completely, but perhaps you could utilize the "downtime activities" outlined in the DMG. His story stays the same, but during the past few months he's learned a few more skills doing XYZ.
I would not allow an arbitrary change that changes stats, unless the adventure is just starting.
However, if the player is insistent, allow it, if the player goes on a quest to find the truth. Like, say their background is that they are an orphan, but they find information that proves his parents are alive. Then, they go on a quest to find them.
But, i would o ly allow changes that are alterable. A child who grew up an orphan in a circus can't relive his childhood and lose the acro bonus.
Maybe a different approach would suit you for your role-playing if you don't want your Soldier's past obliterated by the DM.
You absolutely can change the direction your character goes in an additive manner if you want to spin your role-playing by using Gestalt crossclassing, and it would be very easy to role-play.
Memories from a lost part of your past being awakened or even re-triggered would be more than enough to justify it, if your DM will allow you to play as Gestalt. You can of course get way more creative. Try not to accidentally start role-playing actual game mechanics like possession.
And here's the caveat to both you and the DM:
Firstly, with the halved advancement rate and preference of lesser overlaps, a lot of the raw power is usually drained from the combined class. Further, gestalt characters don’t have an advantage in the most important game currencies: available actions and battlefield presence. Even a character who can fight like a barbarian and cast spells like a sorcerer can’t normally do both in the same round. A gestalt character can’t be in two places at once as two separate characters can be. Gestalt characters who try to fulfill two party roles (melee fighter and spellcaster, for example) find they must split their feat choices, ability score improvements, and gear selection between their two functions.
