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This is correct. To put some concrete numbers on it, I'm in my 6th year at a startup. In 2016, I spent $6,400 to exercise ~50,000 options that had a total FMV of ~$60,000. This is going to add over $7,000 to my 2016 taxes due to AMT.

If you're single and your income is over $115K, or if you're married and your total income is over $150K, there will be tax implications for exercising your options, unless you exercise them when the spread is $0.

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That's not quite how AMT works. AMT is not added to your taxes, it's an entirely alternative tax system (hence the A in AMT). You'll pay the larger of the two values (AMT vs ordinary tax), and given the taxation rates, you may not have made enough to actually trigger AMT. The rate for AMT is lower than ordinary tax, and there's an exemption for the first $N (where N varies based on several factors).

I understand how AMT works; these numbers are from actually calculating my 2016 taxes. With the exercise, I'm paying over $7,000 more than I otherwise would have.

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