When you enter only one form, your income is lower which means your taxable income is lower. When you enter both your 1099-NEC and your W-2 you are increasing your income, so your tax liability would likely increase. Whether or not you are in a higher tax bracket, you have more income that is being taxed. Also, depending on how you are looking at the tax bracket, you may still be in a higher tax bracket when you consider your standard deduction. Meaning, if your AGI puts you in the 12% bracket, it doesn't mean your taxable income would still be in the 12% bracket as your taxable income is lower than your AGI.
For example, if you are single and you had a 1099-NEC with $15,000, your taxable income would be $0 because your standard deduction of $15,750 would wipe out all of your income. If your W-2 was for $15,000 and you had $1,500 withheld from your wages, you would also have taxable income of $0 so your withholdings would be fully refundable meaning you would get a refund of the $1,500 you had withheld. Assuming no estimated tax payments made, you would owe $2,293 for self-employment taxes, but you wouldn't have any ordinary tax liability. When you add the $15,000 from the 1099-NEC to the $15,000 from the W-2, you would have $30,000 total income and $13,107 in taxable income ($30,000- your standard deduction of $15,750- 1/2 of your self employment taxes)
This would give you a tax liability without considering the SE tax of $1,353. Add that to your SE Tax of $2,293, you would have total taxes of $3,646 with a credit for the $1,500 you paid through your W-2 so you would owe $2,146.
Other things that could be happening are removal of certain tax credits when you enter both forms. The Earned Income Tax Credit is one that with less income you would possibly receive, but adding both forms together would make you ineligible.
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