Is there a reason why when I input a 3rd dependent who qualifies as "other dependent" the amount doesn't change/increase.

This is a non-refundable credit. It can reduce or, in some cases, eliminate a tax bill but, the IRS cannot refund the taxpayer any portion of the credit that may be left over

As Affordabletaxes2021 states, the "Other Dependent Credit" is non-refundable and it is likely your income tax has already been eliminated by other deductions/credits on your return.  You can see the calculations in online versions by using the steps below.  

Once the tax on line 22 is reduced to zero by the amount on line 21 no further non-refundable credit is allowed.

1. "Tax Tools" in the left hand menu
2. "Tools"
3. "View Tax Summary" in the main screen
4. "Preview my 1040" in the left hand menu