I use to make good money going after contractors who thought their LLCs would protect them when they vanished with client money. Veil-piercing is actually quite simple for one-man LLCs. As a practical matter, the burden is really on the one-man LLC to prove that the veil should not be pierced, since it's usually pro forma for the plaintiff to show some evidence that the proper formalities and separations were not maintained. (For example, paying for a personal expense like lunch with LLC funds, even once, is enough to shift the burden to the LLC to show that the veil should not be pierced. Not having proper receipts for gas. Attempting to claim a business deduction for part of your garage if you're not charging the LLC market-rate rent. It's a really long list of don'ts. )
Misconduct like "vanishing with client money" is a separate personal tort and/or crime (depending on the specifics) and would make it easy to pierce the veil anywhere regardless of formalities and separations.
That said, for more normal situations that aren't about misconduct, different states take different levels of strictness in analyzing the veil piercing question. YMMV.
But yes, when the proper formalities and separations are maintained, there are many situations where registering and operating through an LLC will limit personal liability.
Example: company signs a contract with no personal guarantee but with a commitment to make 12 monthly payments. The LLC later lacks the funds (contract was signed in good faith). No personal liability.
Disclaimer: I'm not a lawyer, just a law geek layman and former law student. I agree that retaining counsel is the right advice for anything important.