I installed a Leviton 1000W/20A 1HP-120V timer to my bathroom fan, with power going through continuously to a nearby plug. Everything worked fine for about 7 months. Now the fan doesn't work but the electrical plug still works. No lights come on when you press a switch on the Leviton timer. What do you think the problem is?

Leviton 6161 is shown on Leviton as 500 watt incandescent-only 4-level dimmer that cannot control a fan.
If this is a ceiling fan and light, then search for fan light dimmer, or fan control,

If this is bathroom fan-light installation, you are needing a countdown timer rated for both bulb and fan... which will require Neutral wire...
Different timer choices with manuals appear following link:
http://waterheatertimer.org/Countdown-timer-horsepower-ratings.html#Leviton

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Copy following link to identify Leviton timer and download manual
http://waterheatertimer.org/Leviton-timers-and-manuals.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Ada, I think you have already diagnosed the problem. It's the Leviton timer. For whatever reason, it can not handle the fan motor without over heating. The best indicator, was the melted hot wire. Perhaps the switch was defective from the start. I suspect the timer was rated for a ceiling fan and not a whole house fan.

Hope this helps you and thanks for choosing FixYa.

Open following link to identify timer:
http://waterheatertimer.org/Countdown-timer-horsepower-ratings.html#6230

Leviton 6200 series timer replaces single-pole switch. Will not work for 3-way or 4-way applications.
Timer has 4 wires: Blue, Black, White, Green
Timer Green to bare copper wire.
Timer Black to Hot wire from breaker.
Timer White to Neutral wire from breaker. Neutral wires are white wires that are twisted together and covered with wire nut pushed to back of box. If box does not have neutrals, connect Timer white to bare ground wire.
Timer Blue to Load wire.
If timer does not work, reverse Timer Black and Timer Blue.

Countdown timers are made to work with bath fans.
Leviton countdown timers are stout product with 1HP rating.
http://waterheatertimer.org/Countdown-timer-horsepower-ratings.html#6200

It sounds like timer is bad.
Wiring sounds good. Make sure you don't have Load and Line wires reversed. Try reversing Load-Line wires and see what happens.
Another thing that affects electronic timers is line noise/interference. If replacement timer does same thing, then noise is suspect. Try timer on another circuit without fan. Try another brand timer. Update to newer LTB60 Leviton timer. Use cheap circuit tester from home center to check outlet on same circuit.

If this is a new installation, I believe you have a Leviton dimmer instead of a timer. For example, Leviton countdown timers like the 6260 and 6230 have times printed on face, while the similar-looking Leviton dimmer has no times.
Look at instruction sheet that came with product and make sure it says 'timer' and not incandescent 'dimmer'

If this installation worked before and suddenly stopped, then timer has gone bad.
Use circuit analyzer and check nearby outlet to make sure line is clean. Ground faults and circuit noise can adversely affect electronic timers.

Add a comment for more free help.
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on timer or any do-it-yourself project.
Fixya is always less expensive than a service call.

The timer you have purchased will not work for 3-way switching.

We know this because the wires are color-coded.
And you have 2 black wires.
Good job providing that information so I can see what you have.
Both your timer wires are black, and that tells the electrician that the timer is made for single-pole only.

That being said, there are timers that have the same color wires as your 3-way switch, but the timer has to say it works for 3-way circuit.
Usually the instruction sheet or package will say, 'for single-pole.'
Or it will say, 'for single-pole and 3-way.'

1) Aube makes programmable 7-day timers for a 3-way installation.
But this timer is not for countdown.

2) I am searching the web for a 3-way countdown timers, and Levition offers one, but they require you to replace both 3-way switches and then install a 'master-and-remote' combo.
This means each timer is a different model number.
If you want help sorting out that purchase, add a comment and I will research the Levition site.

3) Smarthome sells a countdown timer for 3-way circuits.
It appears to be exactly the product you are wanting.
The product has 95 ratings with 4-1/2 stars
http://www.smarthome.com/2476ST/SwitchLinc-Timer-INSTEON-Countdown-Wall-Switch-Timer-Non-Dimming-White/p.aspx
The product manual is right there on the Smarthome page.
They provide an installation video.
Smarthome is very customer friendly.

I specialize in timers and electrical devices, but I do not know the product shown above.
However I can help you install it.
Add a comment and I will help further.

Try turning off power to the timer for a few minutes, and then turn back on.

If an electronic countdown timer stops working, there are no repairs, and you need another one. Fortunately, inexpensive replacements are available at Lowes, Home Depot and Walmart.
Be sure to mark your wires, or take photo, before removing old timer so new wires connect without confusion.

Switch has 2 black wires. One black is hot wire. One black is wire going to fan. Turn off power. Remove wires from switch and separate. Turn power back on. Using ordinary tester, test each black wire to bare copper ground wire. Only one of the black wires will test hot. The other black wire goes to fan.
Timer black wire connects to black hot
Timer blue wire connects to black wire going to fan
Timer white wire connects to white neutral (in absence of white neutral, connect Timer white wire to bare ground)
Timer green connects to bare ground wire

Only "...smack it with a hammer" if you want to buy a new switch! The weak LED's are probably really neon bulbs, as LED's will either work, or not work and aren't as suseptible to voltage changes as neon bulbs are.

The reason for the slightly dimmer bulbs could be because the plastic lens has gotten dirty, or that the bulbs have moved out of their holders inside the switch.

Provided the switch is still functioning properly, you can disconnect the power to the circuit and remove the switch to see if the bulbs have been dislodged from their mounts, or are lose, etc. Otherwise, if working OK, I'd recommend leaving it be.

If your switch does have the LED's on the side, you might find that the mini circuit board has gotten pushed out of its' proper mounting position and by resetting it back on the faceplace to the proper location, could make the lights brighter again.

Hope you find this Very Helpful and best regards!