No, the 401k at work is one plan - even with the ROTH. While the individual ROTH is a separate plan. The only change would be your income and the limitations imposed by each type of plan. IRS pub 590 shows employer pensions interacting with IRA's and the limitations. A taxpayer can have multiple plans but can't surpass the limitations for each type of plan.
If you scroll down in retirement topics, you will see overall contribution limits to help with your 401k alone.
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