is/are there income limit(s) that restrict the amount of money that a person or couple can invest in a 401k and IRAs?

There are no income limits for being able to contribute to a 401k.

There are income limits for tax deductible contributions to a traditional IRA if you or your spouse are offered a 401k.

There are no income limits for non-deductible contributions to a traditional IRA.

There are income limits for direct contributions to a Roth IRA.

I was discussing this with my Dad and he said that depending on how taxes are filed there can limit where a person cannot max both options out annually, such as $25,000 for 2019. Is this correct?

It's based on your income and whether or not you or your spouse are offered a retirement plan at work (for a traditional IRA).

That being said, the max contributions will be twice as large if you are married filing jointly compared to single.

I assume that you got the $25,000 from $19,000 (401k) plus $6,000 (IRA) and that you are single.

You can max the 401k.

If your MAGI is < $122,000, you can max a Roth IRA. There is a phase out between a MAGI of 122 and 137.

https://www.irs.gov/retirement-plans/amount-of-roth-ira-contributions-that-you-can-make-for-2019If your MAGI is ≤ $64,000, you can contribute up to the max to a traditional IRA and take the full deduction. There is a phase out between a MAGI of 64 and 74.

https://www.irs.gov/retirement-plans/2019-ira-deduction-limits-effect-of-modified-agi-on-deduction-if-you-are-covered-by-a-retirement-plan-at-workNote that you can contribute to one or both types of IRA's up to the max ($6,000 total for 2019).

So if your income allows, you can contribute up to $25,000 between the 401k and IRA for 2019.