This will take some trouble-shooting as there are many project assignments in BQE Core.
Access Project Assignments: View Details for the Phase in BQE Core (and consider viewing details for the Root Project as well in BQE Core) and review the various Assignments.
Open BQE Core> Click on Projects> search for the project or phase> navigate to the far right for a down arrow and choose View Details. Once in the project or phase details, navigate to Settings and then the Assignment tab
1. Review the Rules Here is an example of the Rules selected for this Project:
(In the sample below, these rules are active for this project because of their presence on the list. Click on Add Rule button to see what additional rules could be added. Checking the box to the left of the Description and choosing Delete from the Actions drop-down list allows you to delete rules.)
For example, "Restrict expense entries to budgeted expenses" is an enacted rule for this project (or phase) and that means if a vendor bill in BILL.com is coded to a phase and if the expense on the vendor bill exceeds the phase budget, the vendor bill will not be allowed to post to BQE Core.
The solution is either updating the phase expense budget, changing the phase referenced in Bill.com vendor bill entry, or removing this rule from the phase. 
Click on the [Add Rule] button to see the additional rule options that are available. Here are some of them:
Notice the drop-down box here:
Be sure to investigate Rules, Contacts, Team Members, Activity Items, and Expense Items.
2. Contacts - We haven't seen where Contact assignments impact vendor bills or vendor payments.
3. Review Team Members - Note there are two types of Team Members! Employees and Vendors. If an employee is receiving an expense reimbursement via Accounts Payable or Bill.com, the employee needs to be listed as a vendor Team Member to allow the vendor bill to post to the project in BQE Core.
Here is an example of an employee listed as an employee and a vendor:
Groups - BQE Core allows you to set up Groups of vendors. (Settings> Groups> Create Group button) You can build a group of vendors from the Vendors Contact list. If you have a group for All Vendors, you do not need to add individual vendors.
4. Activity Items - We have not seen where Activity Item assignments impact vendor bills or vendor payments.
5. Review Expense Items- Make sure individual Expense Items or Expense Item Groups are allowed for projects and phases.
Once all of the relevant assignments have been addressed in BQE Core, rerun sync.