I agree
Here are certainly two Fredericks of the right age on the 1861 census.
sharing links
Frederick b 1854 scholar b Ipswich but lived Hadleigh father Edward.
Frederick b 1844 basket maker b Layham a lodger. with Robert Sowman
In 1851 an 8 year old Frederick b 1844, Hadleigh was a lodger with Roger James
Index to the register entry for 23 year old Frederick marrying Mary Rudland in 1867 link
In 1871, Frederick b 1844, a basket.maker b Raydon is with wife Mary and family
Hadleigh, Rayton & Layham are adjacent to each other.
The problem is what happpens to the younger Frederick b 1854 ? He's not with his family in 1871.
Can't find either in the GRO birth index .You might find them with some creative spelling.