The inmates are listed on the 1871/81 Census for that Asylum.
Yes, this was pretty common for asylums. If you are seeking to trace a particular person, it appears some records are held at the Bedfordshire County archives and they have abstracted a few records from around 1880s online:https://bedsarchives.bedford.gov.uk/Disability/MalePatientsInFairfield1880To1882.aspxhttp://bedsarchivescat.bedford.gov.uk/Details/archive/110206813Details of all holdings - would probably need to engage a researcher or pay the archives directly if you can't visit in person).thanks for you reply - I will contact Bedfordshire Archives.