while this is certainly generally true, in at least one specific case a user was able to directly point me to the specific thread and post that got them in trouble, which I had seen before moderators took action, so in that case I have all the available context, and don't see how it deserved a hard warn. (beyond perhaps issues the user had more generally, but it's also a user I know fairly well so I believe I have a good handle on that as well.)
Though I'm not a moderator currently I was back when the site launched, for the first six months or so, and even now still tend to keep an eye on thing going on over on the moderation side of things.
I'm going to avoid speaking on any particular issue or giving specific examples as well but not only have a seen most of the hard and soft warns handed out, I've both handed them out myself and helped train newer staff and reviewed their decisions for approval.
A trend that I've seen is, as Althalus mentioned above, context is often omitted when a user begins to complain about having received action against their account. By this I dont just mean the context of the post and the thread that it is in but also the users history, both recent and long term. I can think of one specific example in fact when a user was given a hard warning and the PM for the hard warning quoted nine different posts of theirs that had broken the CoC in the 48 hours prior to the warning. In addition to this, earlier that week the user had been given two soft warns coupled with PMs detailing what they were doing that was crossing the line and reminding them if they kept doing it that action could escalate. Afterwards the user cherry picked one post of out the nine included in the warning that they felt wasn't bad enough to get them a warning and then began to parade around "look at this post, can you believe they gave me a hard warn for this?" leaving out the fact that they had been pushing the line all week and then finally stepped over it full force with a handful of CoC breaches.
Another thing that tends to be swept to the wayside is repeat offenders. Often time when a user has been given a series of warnings, both soft and hard, for a particular infraction they don't get the same benefit of the doubt as a first time offender. When a mod gives a warning, even a soft warning, a PM is sent to the user detailing what about their post was an infraction and how best to avoid repeating the incident. Even if its just simple advice like urging the user to use the report button and move along rather then engaging in a flame war or responding to baiting posts. Once a user has received multiple warnings for the same sort of behavior, staff tries to engage the person with private conversation in hopes of not only making sure the user understand what they are doing that has warranted action but also talking to the user to find out why they are doing those sorts of things.
As a final note, every user is always given the options to not only discuss the situation with the mod issuing the warning but also to elevate the discussion to include one of the moderator leads. Not all warnings are final and set in stone, and I can vouch that in the past, after discussion on the situation and full context of the situation understood by both sides, action has been lessened or reversed. If you've received a warning from a moderator and feel that it was either undeserved or heavy-handed then inform the moderator that you'd like to discuss it with one of the Moderator Leads. Its not just a tagline to the warning PMs.
TL;DR :
1. Context of a most matters not just looking at what was written in the post itself.
2. User history plays a part. Sometimes people quote what they got warned for, but omit that they've received multiple messages in the past telling them to stop that behavior.
3. Don't agree with a decision? Discuss things with the mod or elevate it to the moderator leads.