Jessie (assume short for Jessica)
No. That is not the case in Scotland, though I believe it may be in England.
The names Janet and Jessie are almost 100% interchangeable in Scotland, and no-one seems to know for certain why this should be. Jessica, as an name in its own right, is extremely rare.
According to Scotland's People there were no baptisms of Jessicas at all until after 1820. From 1820 to 1854 there were 3 in the Church of Scotland and 8 in the Roman Catholic records. From 1855 to 1899 there were just 10 births, though there were 12 Roman Catholic baptisms.
Jessie does occur, but rarely, before the 19th century. Before 1800 there were just 58 baptisms of Jessies, only one of whom was born before 1750, 5 between 1751 and 1779, 22 in the 1780s and 30 in the 1790s.
However there here were 62 burials of Jessies. By way of comparison, for several other names the number of burials is a very small percentage of the number of baptisms. So the use of Jessie for a girl officially baptised Janet does seem to have a long tradition.
Before 1800, however, 141,353 Janets were baptised (not including spelling variants Jannet, Jenet, Jennet, Jonet etc) so the proportion of Janets known as Jessie must be tiny.
Yet almost suddenly in the 19th century, girls baptised Janet turn up in droves as Jessie in later records, and to a lesser extent vice versa.
There are lots of pairs of interchangeable names - Jean/Jane, Donald/Daniel, Peter/Patrick, Helen/Ellen, for example - but in all other cases you can see the logic. With Janet/Jessie, there's no obvious link or derivation.
Digression - There are also some names that ought to be interchangeable, but are regarded as separate names because both turn up in the same families - Jane/Jean and Janet are both the feminine of John, yet you often find a Jean/Jane with a sister Janet/Jessie. Likewise Elspet(h) is originally a variant of Elizabeth, yet you regularly find Elspet(h)s with a sister Elizabeth. Isabella is the Spanish version of Elizabeth, yet you find families with both an Elizabeth and an Isabella.
For fuller information see
www.whatsinaname.net