It is possible, but the process to switch builders after you close (and before completion) can be quite time consuming. The most common reasons homeowners want to switch is due to disagreements on costs and excessive delays.
When a homeowner wants to change builders, both the lender and the title company will have certain requirements. The title company will generally require the original builder to provide a final lien waiver stating they are owed no additional funds. When there is a disagreement of costs, that can be difficult t obtain from the builder. The homeowner will want to make sure all disagreements regarding costs have been worked out with the original builder. Without the final lien waiver, the project may be halted until one has been obtained.
The lender will want to review the new paperwork from the builder- a signed contract and budget. If the budget has increased, the homeowner will need to bring in more funds before the project can continue.
So as you can see, it is very important to make sure the builder you decide to work with is on the same page. Speak with past clients of previous projects to see what their experience was and if there were any issues completing on budget. Be sure to have a full understanding and expectation of the timeframe it will take to complete the project. Some would recommend even writing the time frame into the contract.