Do carbon resistors not follow tolerance as they age?
So you are using 5% resistors and are worried about stability over time.

Your 1kΩ, 5% resistor can range from 950Ω to 1050Ω.  There is no substitute for getting a good quality DMM and making appropriate measurements.

From Resistor01:


  Shelf Stability/Drift. In addition to all the environmental and electrical factors listed above, resistors experience a long-term drift that may be as high as 2% per year for carbon composition resistors. Metal, oxide, metal film, and wire-wound resistor are
  typically 10 times more stable. 




So carbon resistors can change ±2,000 ppm/year, which would equate to ±2Ω for a 1kΩ (0.2%, which is not 2%).

From Carbon composition resistor:


  Fifty years ago, carbon composition resistors were widely used in consumer electronics. Because of the low stability of the resistance value, this type of resistor is not suitable for any modern high precision application. For example, the resistance value can change up to 5% over a shelf life of one year.  With heavy use the value changes even more: up to 15% for a 2000h test at full rating with 70°C. Soldering can cause a 2% change. The reason for this instability is inherent to the design of the resistor. The carbon composition contains materials with different heat expansion properties. When the conducting carbon particles and the nonconducting binder heat up or cool down, stresses arise in the resistor body. The mechanical contact between the conducting particles will change, and this leads to a change in resistance value. Also noise properties are not good due to the mixture of different materials. The noise level increases when current flows. Resistors of 0.25W and 0.5W, have a maximum voltage of respectively 150V and  500V. Insulation resistance is poor with approximately 10^9 Ω (order of magnitude worse than other types). One more reason for the decrease in the use of this type of resistor, is the high temperature coefficient of around 1200 ppm/°C. The operating temperature range is between around -40 to 150°C. However, the resistor derate above 70°C.


Proving you get what you pay for!  If it is critical maybe a better quality resistor is required.

If something is that critical, you have to get into the datasheets and measurements.