The simple answer is 'Yes'. Whilst a lot of home movers would prefer to use the same solicitor for both their sale and purchase, there are certain scenarios where you may need or choose to use separate solicitors for your sale and purchase:
Conflict of interest
You may be in a situation where the people you are buying from may be using the same solicitor. In which case, if your seller is not willing to source a different solicitor, you would need to find another solicitor to act on your behalf. However, that doesn't mean that you cannot use your preferred solicitor for your own sale.
Incentives
You may be selling your house through an Estate Agent who are offering incentives such as a discount or cash back if you use their preferred solicitor. Whilst it may make financial sense, you may still want to use your preferred solicitor for your onward purchase.
Geographical area
Some conveyancing solicitors may only work around the local geographic area, so if you're selling in one town and buying in another, you may wish to use a different solicitor for your onward purchase.