According to wikipedia, CAT 6 standard:
Category 6 cable, commonly referred to as Cat 6, is a cable standard
for Gigabit Ethernet and other network physical layers that is
backward compatible with the Category 5/5e and Category 3 cable
standards. Compared with Cat 5 and Cat 5e, Cat 6 features more
stringent specifications for crosstalk and system noise. The cable
standard provides performance of up to 250 MHz and is suitable for
10BASE-T, 100BASE-TX (Fast Ethernet), 1000BASE-T/1000BASE-TX (Gigabit
Ethernet) and 10GBASE-T (10-Gigabit Ethernet).
And Cat 5e:
The specification for Category 5 cable was defined in
ANSI/TIA/EIA-568-A, with clarification in TSB-95.
These documents specified performance characteristics and test
requirements for frequencies of up to 100 MHz.
Assuming these premisses, you can see that the main phisical reason to addopt a Cat 6/6a wire and conectors is the possibility to use more than 100 MHz to send/receive data signals, what you will actually ned only on Gigabit speed(such as 1000BaseTX Gigabit ethernet).
Adding Cat 6/6e conectors on a Cat 5e wire network will only provide the capability to use up to 250MHz into your conectors, what wouldn't be noticeable in terms of speed on any way(unless your Cat 5E conectors are bad) because the main device on a wired network is acutally the wire, not the connectors. Actually, if you had a Cat 6 wire network using Cat 5e conectors then you could maybe see your Gigabit Ethernet running Ok(Ok not means good or perfect, you could see a lot of packet loss).
If you want to use a Cat 6/6a network, then you should use all components of this standard(cables, connectors and actives) and not only one of this component alone.