I have 2 DHT11's connected in front of me on a neat breakout cabled breadboard from my Uno with 2 DHT11's connected with the recommended pull-up separate for each sensor and a 100nF capacitor across each sensor Vcc and ground lines, both have been powered up and working interchangeably overnight and over day too... as I wanted to see if there was any drift in the devices. Both have worked for several days now without any issues whatsoever except possible calibration.
So since you didn't post your code I am having to get out my crystal ball and suggest that it isn't wired properly, either you are missing the pull-up or you are missing the capacitor that the data sheet and the sketches and other material found here recommend, By here I will point out that there are several articles/examples on this topic on this site and this forum topic specifically, that show how to and why you must connect the sensor to the Arduino in a specific manner and method. Failure to observe these basic steps frequently leads to issues regarding erratic or non operation of the Device/Arduino/Sensor however it will make you happy to know that it takes real work to do any of the above and successfully "Brick" either the Sensor or the Arduino, and the Arduino can be repaired for around $5.00, the cost of a new chip W/bootloader and they plug in. If you change the wiring to the Recommended connections/parts and it still fails then you might re assign the DHT11 pin to another pin in the sketch and try again. Should the results be the same then surely you have either a bad sensor or your wiring is still not correct. My Sketch worked great from the beginning, after I got a lesson about where to put things,... no, not there... libraries... I then modified the\ sketch to talk to a serial display and removed the temperature measurement function and have had the thing work constantly Since. I an a complete NOOB to Arduino but I am a skilled electronics oriented person and at age 66 (in june) I have learned to follow instructions... Sooner or Later... usually later as I am as hardheaded and obstinate as they come... and I write long and really boring replies,,, Dementia setting in, I suppose. IMO Hope this was more helpful than insulting, It wasn't meant to be insulting but you can never tell about us Old Geezers... (nor can we)
Doc