Sure. But the problem bingybear is aluding to is that if you have a TIRA from your 403b with zero basis and then do a backdoor Roth by making a non-deductible TIRA, then your subsequent conversion to Roth is considered to be taken pro-rata from both portions of your TIRA (or both TIRAs if you're using separate accounts).
Suppose you have $95K in your 403b-originated TIRA and you contribute $5K to another TIRA and then convert that $5K to a Roth.
What you think you're doing is moving $5K from that second TIRA into your Roth. But the IRS looks at it as one virtual $100K TIRA and that you converted $4,750 (5% of $95K) from your 403b-originated TIRA and converted $250 (5% of $5K) of your second TIRA.
And unfortunately the law doesn't give you the choice, and the law doesn't care which TIRA account the funds actually came from.
Further unfortunately, if you've been doing this for several years, I'm not sure if there's any clean way to undo it. If you want to do things properly, you probably need to go back and amend the past several years of income taxes to reflect things accurately. You'll at least need to file Form 8606 for the applicable years, as well as amend your federal income taxes, because you'll owe ordinary income tax on the portion of the conversion that was attributed to the 403b portion - so in the example above you'd owe ordinary income taxes on an additional $4,750 of income.
A second effect which you'll need to know is that your basis in your Roth is different between a clean backdoor Roth and a pro-rata backdoor Roth that you evidently have done. You can google "pro-rata Roth conversion" and probably get some more information.
I'm sad to be the bearer of bad news.